This custom is mentioned explicitly in the Rema, OC 299:
וי"א לדלות מים בכל מוצאי שבת כי בארה של מרים סובב כל מוצאי שבת כל הבארות ומי שפוגע בו וישתה ממנו יתרפא מכל תחלואיו [כל בו] ולא ראיתי למנהג זה
And there are those who say to draw water from wells or natural springs every Saturday night, because the well of Miriam passes through all the wells and natural springs every Saturday night. And one who touches it and drinks from it will be healed from all his afflictions [Kol Bo]. And I did not see (anything) about this custom.
As has already been noted this point is also raised in the Shulchan Aruch HaRav 299:20 as well as the Beis Yosef 299:13.
Interestingly, I came across a sicha of the Lubavitcher Rebbe here, Motzei Shabbos, Parshas Chukas (1979) תשל"ט which perhaps can shed some light as to whether this practise has any relevance today, especially given the fact that the Rema says "ולא ראיתי למנהג זה" - "And I did not see anything about this custom."
In short, the sicha mentions that at first glance, the fact that the Rema says ולא ראיתי could imply that one should not do it. However, the Rebbe counters that conceivably the reason why the Rema said he never saw it was in order to prevent the women in Cracow, Poland from running immediately after barchu to draw water from springs and wells. As such, it does require further examination. He mentions that the Shulchan Aruch HaRav who came after the Rema (as mentioned above) brings this idea but does not say like the Rema that he never saw this idea explicitly. As such he writes:
אפשר לומר על זה לכאורה שזה דבר שנשאר באמיתיות, שלמנהג זה יש מקום גם היום, ולראיה מביא זאת אדמוה”ז בשו”ע שלו!
It is possible to say about this, at first glance, that this (custom) remains true, that this custom has a place even today, and the proof is the fact that the (Alter) Rebbe (Rav Shneur Zalman of Liadi, the Shulchan Aruch HaRav) brings it down in his own Shulchan Aruch.
And the Alter Rebbe made his Shulchan Aruch “השווה לכל נפש” - "to be equal to every soul" i.e. accessible for everyone, which would mean it is still applicable today. He continues, that if we focus on the reason behind the custom, namely that it results in a refuah (a cure) for those who need, then "המעשה הוא העיקר" - "the action is the main thing", and as such:
כאשר מביאים בהלכה סיפור מעשה אז זו ראיה מוכחת שההלכה היא הלכה קיימת!
When you bring a story of action into Halacha, then it is proven proof that Halacha is existing Halacha!
He then qualifies the Kol Bo (the source of the custom) by saying that there it also clearly spells it out as a מעשה, an action - i.e. that a woman would go to pump water and if she delayed her husband would be upset or 'angry' as per the expression used in the Kol Bo. And out of fear of incurring her husband's wrath, the pitcher fell and broke, and the water she drew spilt out, and thus (I'm not entirely sure what this means) she really drew from Miriam's well, and her husband had a p'sak (a halachic ruling) as a result of the water splashing him. In those places where they hurt his body, they were healed with complete healing, and in the other places where the water did not reach, he remained, with his health opposite to what it was before.
Explains the Rebbe, it is a matter that brings a cure and health and justifies the Gemara in masechata Shabbos 35a that if one want to see Miriam's well he should ascend Mount Carmel and he will see the sea of Tiveria and then a rock that resembles a sieve - and that is the well of Miriam.
He continues that this Gemara is somewhat cryptic, but he reasons that is relevant to the story related in the Kol Bo. In addition, Rav Chaim Vital relates that he sailed with the Arizal in the sea of Tiveria, and the Arizal drew water when they came to a fixed place, and commanded him to drink, and said "You drink from Miriam's well."
As such, the Rebbe concludes that the Well of Miriam clearly works in a physical capacity, like the Kol Bo seems to suggest, which means all the more so, it works spiritually. Therefore:
מובן שברוחניות בוודאי שהמנהג קיים, ו”לא ראיתי מנהג זה” אפשר לומר רק בנוגע לגשמיות הענינים, לגשמיות המנהג
It can be understood that spiritually the custom definitely exists. And "I didn't see this custom" (as stated by the Rema) can mean only in a physical sense of the custom.
And when one learns from what the Shulchan Aruch HaRav wrote on the matter, we can assume that the healing nature of Miriam's well continues every motzei Shabbos and lasts all week, allowing for a healthy week in both physicality and spirituality, as well as the study of Torah and mitzvah observance.
In short, he thinks it is still very much relevant today, and there is nothing to be lost from doing it.