In present day world, most of the democratic countries follow Secularism, i.e. separating state and religion. What is the opinion of Judaism on Secularism: Should a state following Halacha endorse Secularism? Should such a state keep religious matters separate from government?
First of all, it cannot be emphasized enough that Judaism does not, and certainly since the days of the Sanhedrin, has not had, an official spokesman. Accordingly, it is essentially impossible to answer what Judaism thinks about this; only what individual Jewish scholars, basing themselves off, among other things, Jewish teachings, have held.
That said, below are a couple of rabbbis' views on the relative merits of a secular democratic state, in which religious institutions are not granted authority over the masses, vs. a theocratic state.
Here is what R. Shakh wrote about democracy (Mikhtavim u-Ma’amarim, vol. 5, p. 124):
בל נחשוב, שהשיטה הנקראת "דמוקרטיה" היא דבר חיובי . . . האמת היא שהיא אסון לעולם. היא נותנת הרגשה מדומה של "חופש" בו בזמן שלאמיתו של דבר היא רק הפקר, ותו לא . . . הדמוקרטיה היא דבר טרף, וכל כוונתם לעקור דרכה של עם ישראל ולהרסו
That is: Do not think that the system called "democracy" is a positive thing... The truth is it a disaster to the world. It gives a false sense of "freedom" which while in fact it is just anarchy, nothing more. . . Democracy is something non-kosher, and their whole intention is to uproot to the path of the people of Israel and destroy it.
On p. 127 he writes:
ואנו תפילה להרבונו של עולם, אנא פטור אותנו מקללת הדמוקרטיה החדשה שנשלחה לעולם, שהיא ממש כמו מחלת הסרטן שנשלחה לעולם. כי רק התורה הקדושה היא הדמוקרטיה האמיתית.
And we pray to the Creator of the world, please rid us of the curse of the new democracy that was sent into the world, which is just like cancer which has been sent to the world. For only the Holy Torah is true democracy.
here is a passage, from R. Yissachar Meir, that appeared in an official Degel ha-Torah publication, Ve-Zarah ha-Shemesh (Bnei Brak, 1990), p. 630:
טעות אחת טעו מנהיגיה הראשונים של המדינה, הם חוקקו חוק הנקרא "דמוקרטיה". כל אחד יודע דמוקרטיה זו מהי, על פי השיכורים הנמצאים במדינה – שלוש מאות אלף מסוממים חיים במדינה – ועל פי זקנים מסוידים וכו' נקבע השלטון. כמו כן בכל מיני שוחד, ודרכי כפיה, נקבע ע"י מה שנקרא "בחירות", איך תנהג המדינה בכל הנושאים העולים על הפרק. על פי דרך התורה, גדולי התורה הם הקובעים את המנהיגות.
One wrong mistake the first leaders of the country, they passed a law called "democracy". Anyone know what the democracy is, according to the drunks who are in the country - three hundred thousand drug addicts live in - and according to old painted sheet set rule. Also all kinds of bribery and coercion methods, determined by the so-called "elections", how the state will handle all matters arising on the agenda. According to the way of the Torah, the Torah sages take precedence leadership.
R. Elhanan Wasserman writes in his Ikveta di-Meshiha, par. 2, published on the eve of the Holocaust.
הגיע כבר העת שתבינו, כי בלעדי אין מושיע. אבל העם מסרבים להבין. עוד נאחזים בשולי הדמוקרטיה הגוססת. אף היא לא תועיל, בדומה לעבודות הזרות הקודמות.
It's already time to understand that without me (God) there is no savior. But people refuse to understand. More cling to the edges of the dying democracy. Although it will not help, just like the former idolatries.
All of the above quotes are found here. For a discussion of rabbis who were supportive of communism, see here. This is significant since a communist state would be equipped to impose religious (or any other standards) on its citizens. This also demonstrates the degree of possible divergence of views within "Judaism".
Although rabbis Shakh and Wasserman, et al were opposed to Secularism, R. Schlesinger decided to set up his own system. This system he outlined his book Kollel HaIvirim. He explains his system would be democratic. He explains that the Torah requires one to follow the majority. This is so, even when a Godol or the like holds different views. He proves this by pointing to the system of the Sanhedrin. There, they did not just go with greatest Rabbi on the Sanhedrin, rather, they started polling the views of the lowest one. (Source)
This system seems much more democratic, and thus closer to secularism.
Additionally, R. Hayyim Hirschenson had positive feelings about democracy.
Whatever secularism we endorse is due to some temporary benefit of this over its alternative. For example, if a secular Muslim is less dangerous than a religious one, we will endorse that. It all depends on the situation.
The roots of evil such as jealousy, lust, and honor (avot 4:28) are not cured by secularism so we cannot give a general rule here of what's better than what. History shows all the philosophies out there eventually lead to anti-semitism anyways, so we have to play it by ear.
Ultimately, the Jewish view is to endorse only the torah way as we say three times a day in our prayers (Aleinu)
Therefore we put our hope in You, Adonai our God, to soon see the glory of Your strength, to remove all idols from the Earth, and to completely cut off all false gods; to repair the world to the Kingdom of God. And for all living flesh to call Your name, and for all the wicked of the Earth to turn to You. May all the world's inhabitants recognize and know that to You every knee must bend and every tongue must swear loyalty. Before You, Adonai, our God, may all bow down, and give honor to Your precious name, and may all take upon themselves the yoke of Your rule. And may You reign over them soon and forever and always...