There are two matters here that are quite separate but related.
Monasticism as practiced by the Christian church started at least 100 years after Christ and comes in numerous "colours and flavours". In its extreme form it shows itself as orders like the Trappist Monks, to completely dispersed orders that have no institutions at all. There is everything in between.
Originally monasticism comes from the Greek word μοναχός (monachos) which derives from the root word μόνος (monos) meaning 'alone'. Thus, pure, original monasticism meant monks were supposed to stay separated from society and devote themselves to humility, separation, prayer and mortification of the flesh. Later, some churchs added the requirement of chastity and/or poverty. Indeed, Col 2:23 speaks AGAINST such orders as follows:
Such restrictions indeed have an appearance of wisdom, with their
self-prescribed worship, their false humility, and their harsh
treatment of the body; but they are of no value against the indulgence
of the flesh.
The word (correctly) translated "humility" here is ταπεινοφροσύνη (tapeinophrosune) is rendered (inexplicably) "asceticism" in the ESV for which it is alone.
The problem here is the "self-prescribed worship" and the false humility, done, presumably for show (ironically!!). Paul specifically says that such practices do not work. There is no scriptural basis for monastic orders.
Now, having said that, there are some good points about monasticism that cannot be ignored. Not all "monasteries" are intended to keep their inhabitants separate, isolated and in useless pursuits. Indeed some orders are dispersed, or, encourage the public to participate in their work. Here are some spectacular examples that no one would object to:
- Schools - all schools before the 17th or 18th centuries were run and staffed by an order of monks who taught. Over time, the monks struggled with numbers and started hiring non-ordained teachers. In the 19th century, schools started being founded by various secular states but many "church" schools remain.
- Universities - all universities until the 17th or 18th centuries (the transition varied a lot by country) were rub by ordained scholars. There is the famous example of Sir Isaac Newton, an Arian, who had to be ordained to take the Lucian chair of Physics at Cambridge, so an exception was made that, despite the rules of the Anglican church, an Arian was ordained to the priesthood.
- Hospitals - there is a similar situation with hospitals as well. Even today, many hospitals are operated by various church based denominations.
I am sure others will think of more examples.
Col 2:23 is speaking AGAINST separated monasticism. It says nothing (for or against) about profitable institutions such as universities and hospitals, missionary orders, Bible societies and many other groups that in numerous ways fulfil the mission of Christ as set out in Matt 28:19,, 20 -
Therefore go and make disciples of all nations, baptizing them in the
name of the Father, and of the Son, and of the Holy Spirit, and
teaching them to obey all that I have commanded you. And surely I am
with you always, even to the end of the age.