NOTE: This answer brings a commonly held opinion, and should be treated as a non-comprehensive, non-rigorous introduction (for example, it does not offer a rigorous definition of "the public"), and provide some sources to get started. There are many essays and responsa on these topics that go into more detail, and fill books. As MY is not a debate site, it would be difficult to cover every possible point and pre-empt every possible question, although feel free to ask and I will answer some questions, mainly just to demonstrate that there are answers, and encourage the reader to go and do their own further research into the matter.
One strong and common opinion is that yes it is permitted, some say obligatory, and certainly strongly urged. It is what will bring the Moshiach to our very low generation, who need "a very great light". However, there is a litany of cautions and guidelines and limitations, and one must follow them. Some of them are collected below. However, let's start with the sources:
Sources
Collected from my own learning and verification where I have access, and the verification of others where not.
Eitz Chaim, the major Kabbalistic work of the teachings of the Arizal by his foremost student, R' Chaim Vital T'zl, Shaar HaHakdamot where he writes that ideally one should learn Talmud first, and then move on to Kabbalah. Everyone should do so; anyone who doesn't causes sorrow to the Divine and delays the Redemption. He adds that if a person has issues with learning Talmud, they should go straight to the Kabbalah. He brings many sources for this, disagreeing with the "only after 40" idiom (see Shach, Yoreh De'ah 246:6).
Even Shleima by the Vilna Gaon T'zl, 8:24 and 11:3, as well as his commentary on Mishlei 4:22, 5:18, 7:12 and 20:9, and on Tikunei Zohar 107:3 (plus many other places). In his commentary on parshat pikudei, 17 and Mishlei 2:9 he writes that it is obligatory for everyone to learn Kabbalah1. His position amounts to the fact that the learning of Kabbalah will be the primary cause of the Redemption. In Kol HaTor he predicted that in the year 5750 (1990) the "revelation of Kabbalah" will begin.
Ohr HaChama by R' Avraham Azulai T'zl writes in the introduction that the decree of not openly engaging in Kabbalah was lifted in the year 5250 (1490)2, and in 5300 (1540) it became praiseworthy to do so in great numbers in order to bring Moshiach.
HaSulam by Baal HaSulam T'zl, where he writes in the introduction that the time has come for everyone to learn Kabbalah1.
Eder HaYaker v'Ikvei HaTzon, p.144 R' Kook T'zl explains that it's no longer appropriate for the "big questions" to only be answered to the great Rabbonim, but everyone should have an answer, and all of klal Yisrael should participate in this process.3.
The letter of R' Yisrael Ba'al Shem Tov T'zl to his brother in-law, Rebbe Gershon of Kitov T'zl. In it he states that the Moshiach will come when the wellsprings of his teachings have spread [and been taught] to the whole world. The teachings, which he states he wasn't permitted to give in his own lifetime, are Kabbalistic in nature.
The Zohar itself, in several places explains how the teachings of Kabbalah will bring the Redemption. See 1:117a, taken to mean that in the year 5600 (1840) the wellsprings will burst forth from above and below, meaning Torah wisdom that was hidden will become available (Kabbalah, note the same word used in the Ba'al Shem Tov's letter), and this will coincide with a great influx of "lower" wisdom. See also 3:124b which states that specifically the teachings of the book of the Zohar (a primarily Kabbalistic work) will bring about the Redemption1!
[I believe that R' Chaim of Volozhin T'zl, before he was niftar, said that it's extremely important that the entirety of his work Nefesh Hachaim be learned by everyone. It is a work that contains a lot of Kabbalah, which is written in such a way that clearly demands there be a teacher to explain it. I will look for that source again bli neder.]
Dangers
There are several noted dangers about teaching Kabbalah that cause us to hesitate and make sure we do so wisely. Here are just a few:
It can be misunderstood. Chacham Hillel Shlita, in Faith and Folly, goes into much detail about this danger. Two of the greater issues are:
- One might misunderstand the metaphors used and consider Hashem to be in some way composed of parts, chas veshalom.
- One might come to consider that Hashem has partners on His level, chas veshalom.
Kabbalah is a deep, mystical subject that deepens one's knowledge of Hashem, and misunderstanding it can lead to misunderstanding Hashem - Who wishes to be known - a great sin. Much of the original writings are written in a form of linguistic code, that requires a cipher to understand. If it is not learned with the explanations of the Tzaddikim that have come since then, then one has no hope of understanding anything, and is in great danger of misunderstanding something and falling into blasphemous thoughts, chas veshalom.
It can become profaned. These are very holy, personal, and even private (mature) teachings, about Hashem Himself, and should always be treated with extreme awe and trepidation. By exposing these teachings to "popular" culture, they can (and unfortunately have in many cases - "Madonna grade Kabbalah") become profaned.
It can put people off. See the story of the Splattered Gem by R' Pinchas of Koretz T'zl and explanations. This is a three fold issue.
- If it taught to someone not ready or through a faulty teaching or inexperienced teacher, it might put them off Torah (or worse, come to disdain it) rather than bring them closer, which is a huge tragedy and must be avoided.
- These teachings are considered some the highest teachings of all, and lay out openly the secrets inside the Torah. After hearing them, if a person continues to sin afterwards, he has no further excuses or mitigating factors of "he couldn't have known better".
- In addition to the last point, if a person is unmoved by these teachings, then it is considered a tremendous waste. If the deepest truths about Hashem do not move someone, then what will? Those teachings have been wasted, and that soul has little hope.
Kabbalah itself is dangerous. Kabbalah is a very powerful wisdom, and can be misused, and certainly should not be available to malevolent sinners who would do so. See the aforementioned Faith and Folly for much detail about this. It has also been the cause of several disasters in our history, such as the Shabtai Tzvi false messiah tragedy.
Another commonly cautioned issue is it can confuse and distract. Perhaps even make one go loopy (see the story of the Pardes in Chagiga 14b). It can distract one's focus away from learning halacha, which is an absolutely critical part of a persons' service.
Guidelines and Limitations
It goes without saying that one should not teach what one does not know very well. Therefore, before teaching anything, one should have discussed it properly with one's own Rabbis (who themselves obviously need to be experts in these topics), who should offer feedback, including as to whether one is actually good at explaining and teaching these things. There is a difference in being able to recite the sefirot and olamot and their connections, and actually understanding that and being able to translate it into a plain language every-day lesson. If one can't answer intelligent questions on a subject, they probably don't know it well enough to teach it - this is true of any subject.
One should not teach "pure Kabbalah". Faith and Folly addresses this, and as I haven't learned any "pure Kabbalah", I can't explain what it is. However, there is much "distilled"4 Kabbalah available nowadays, and there are many sefarim aimed at the public that contain a lot of Kabbalah, including works by the Maharal T'zl, the Ramchal T'zl , R' Chaim of Volozhin T'zl the Chassidic Rebbeim. There are contemporary works by well respected Rabbonim such as R' Aryeh Kaplan T'zl, R' Yitzchak Ginsburg Shlita, R' Mattis Weinberg Shlita, R' Moshe Miller T'zl, R' Manis Friedman Shlita, R' Akiva Tatz Shlita and others. If one wishes to teach Kabbalah to the public, one should follow their guidance and use their explanations and not stray further without asking one's own Rav.
The Ba'al Shem Tov has stated one shouldn't teach Kabbalah without his teachings (which later became known as Chassidus). See Derech Mitzvotecha, Shoresh Mitzvat ha-Tefillah 2 by the Lubavitcher Rebbe.
One should know one's audience. Ideally, one shouldn't teach it to people who haven't got a good basis in Talmud and Halacha5, as well as trained in how to think and reason, as cautioned by the Ramchal. Where this is not possible, one should teach Kabbalah in a very distilled form, free of jargon, with very well thought out parables and plain, every-day explanations following after the great teachers of Kabbalah outlined in points 1 and 2, and make sure to include disclaimers about the dangers outlined above.
One should become well acquainted with the dangers, and their common causes. Now that all Kabbalistic works are available publicly online, and in popular books, there are a lot of questions and there is a lot of potential for misunderstanding, so it is important to know how to answer some of the main misconceptions, and offer some explanations to those who read these works and have questions, so they don't err.
As with all hashkafa, one should always bring the lessons back to the practical. Their main goal is to inspire and bring people genuinely closer to Hashem and thus to enhance people's adherence to halacha, the performance of Mitzvot, prayer and chesed.
In general, don't do it alone. Just like one shouldn't learn it alone, one shouldn't teach it alone, but with guidance from one's teachers, and without straying from the words of the Tzaddikim who have, themselves, written teachings of Kabbalah designed for the public.
Conclusions
It is indeed the case that we have a strong historical tradition that claims teaching Kabbalah openly is a grave sin. However, we also have a strong tradition of permitting it, with explanations of why this has changed (or that this change was always intended), along with guidance, limitations and rules. We shouldn't ignore this, nor the fact that many great Tzaddikim for at least half a millennia have poured their hearts and souls into finding ways to distil essential lessons from Kabbalah so that the masses can understand them. With the Zohar itself claiming that our modern generation is nearly 2 centuries into an era of a great upsurge in holy and secular wisdom, which go hand in hand, and claiming that learning and teaching it is a critical aspect of bringing about the Redemption, we certainly should take this to heart.
However, as I opened the conclusion, this is indeed not the final say, and is certainly a highly charged issue, and there is strong disagreement on many aspects. Therefore as always with answers on this site, one shouldn't take this as their final say and charge, but should speak with ones own Rabbi and find out what one's own tradition says on the matter, and follow it, as well as taking into account all the answers to the question, not just this one. I am not writing as a "Rabbi" in authority, but as a student myself, of my own Rabbis who have their own traditions on this matter, which I strive to follow without error.
1 - It is logical to assume, as many have, that once the public is allowed/obligated to learn it, the learned are allowed/obligated to teach it, as it is dangerous/impossible to learn Kabbalah by oneself, and it is generally bizarre to set up a system where people can learn something that nobody is allowed to teach. This answer does not deal with whether it should be taught actively or passively or in between.
2 - It is not clear if this is overruling the Mishna in Chagiga or not, if the Mishna's ruling is absolute or not, or if the Gemara opens nuance to the Mishna's statements. To get an answer to this, one should look for the responsa of the Tzaddikim on these matters, according to one's minhag.
3 - See also Igerot haRa'ayah, vol. 1, Letter 43. He was a proponent of demystifying Kabbalah and introducing it to yeshiva curriculums.
4 - Not distant quasi Kabbalah stuff, as one comment has asked. If the totality of the answer is in mind, it's clear this word does not make the opening paragraph "empty". Distilling means a) getting to the heart of a particular message in Kabbalah and b) finding a way to explain it without jargon, through mashal, in a way that the every day person can absorb it. The essence is very much preserved. The Tzaddikim have given us many examples of this, especially Chassidus, that we can borrow, and this late in the game, much has been distilled. There are different degrees to which things are distilled and much to debate about this, including what prohibitions still exist in which areas.
5 - This has always been the case, the difference now being that everyone is now able to learn the Kabbalah, rather than a select few. It is still lechatchila to have this grounding. Debate can be had about how much Talmud and Halacha counts as "being well grounded"