This is such a cool question, but it's rather complicated to answer and it also a pretty controversial topic, particularly amongst religious non-drinking Muslims. I also generally focus on other things, but I’ve found several references to alcohol and alcohol consumption in my studies, so I’m gonna take a shot at it either way. I'm pretty sure someone knows much more about it than me, though.
The short answer is: the state generally didn’t really care who was consuming what. The long answer is below:
Firstly, there is no such thing as ‘The Islamic Civilization’. Already directly following the Islamic conquest you notice that Islamic societies take on an extremely diverse character, and what may have been practiced in Morocco could well be considered heresy in Iraq, or vice versa. This also directly affected the way different rulers interacted with things nominally considered forbidden or taboo, such as the (public) consumption of alcohol.
Either way the production and consumption of alcohol in Islamic societies, particularly amongst the elite, goes back pretty much until the founding of Islam itself. I’m going to look at several different things, firstly, about what the Islamic tradition itself says about the consumption of alcohol, then I’m going to look at how alcohol is portrayed in the literary sources, and then I’m going to discuss what we actually know about its consumption.
It is generally held that the Qurʾān forbids alcohol outright. This is definitely a position found amongst many Muslims nowadays, but the phrasing itself is ambiguous enough to warrant a degree of interpretation. In any case, the Qurʾānic citations were not considered an immediate prohibition (not in the earliest phases, at any rate). Moreover, the Qurʾān also refers to “rivers of wine” in its description of Paradise, so ḫamr was not necessarily seen as a bad thing per se. In the ḥadīth literature, however, the tone is dominantly negative: I just googled for ḥadīts on alcohol (in Arabic), and they all condemn all sorts of alcohol consumption, even the smallest quantities.
However, as Roger Allen pointed out: “there is considerable debate as to the precise definition of ḫamr”, and it is from the earliest moments that we know of disagreement within the Islamic community regarding to what could and could not be considered ḫamr. Some people brought up the issue of spirits, which was considered to be the same as ḫamr by all legal schools, except the Ḥanafis. In any case, there were many long discussions about the exact definitions of what kinds of drink fall under the (post-)Qurʾānic ban on ḫamr, but as we all know, there are generally differences between what is proscribed and what people actually do.
We know that people throughout the Islamic world consumed alcohol. How do we know this? Well, firstly, there’s a good deal of Arabic poetry that deals directly with the consumption of wine. A particularly fine example of wine poetry was composed by the third Umayyad caliph, al-Walīd b. Yazīd (translation Roger Allen):
Cast off hidden cares with frivolity; thwart fate by enjoying the daughter of the grape
How I long to drink from a maid of noble descent on her wedding-day
Resplendent in her jewels, wondrous to behold
As though her glass contained a firebrand gleaming in her eye
There’s a really long article on ḫamriyyāt poetry in the Encyclopedia of Islam as well, and the aforementioned Allen also spends some time talking about it in his Introduction to Arabic Literature. Secondly, the 15th century Arabic-language manuscripts of the Arabian Nights contains a quaint little story of the Porter and the Three Maids, who embark on a raunchy sex adventure facilitated by a copious amount of alcohol consumption. They are later joined by the Caliph Harūn al-Rašīd (of Civilization IV fame), which tells us something about how alcohol was perceived by the general population, generally as something awesome.
In any case, the consumption of wine is attested for more or less every period in Islamic history: it got a particular importance amongst Sūfī orders from the 13th century onward, the idea being that the consumption of alcohol and the resulting stupor could bring them closer to God. It is sometimes said that the wine-verses of the Sufi poets must be taken metaphorically, and there is definitely a lot of mystical subtext in Sufi poetry, but considering the large scale on which alcohol was being consumed, this is probably more an indication of historical whitewashing than anything else. In any case, here’s a piece of poetry from the 13th century mystic ʿUmar b. al-Farīd (d. 1235) (translation Allen):
In remembrance of the beloved we drank a wine through which we became drunk before the vine was ever created
For a cup it has a full moon; it is a sun circle by a new moon; when it is mixed, how many a star appears!
So people were consuming wine, but where did they get it from? In all fairness, even nowadays, it’s not so hard to get wine in Tehran (especially if you’re not a Muslim), as local Christians and Jews are allowed to make it. We don’t really have that much information on what the process of wine production looked like, but it appears that in terms of legislature, restrictions on the production of alcohol were rather lax: as far as we know, the production of alcohol was not illegal, and there are various handbooks detailing what kinds of ingredients were necessary for making it.
References
Allen, R. An Introduction to Arabic Literature. Cambridge: Cambridge University Press, 2000.
Hadawi, H. The Arabian Nights. New York: Norton, 2008.
Irwin, R. The Arabian Nights – A Companion. New York: Tauris, 2004.
Sadan, J. “Khamr”. In Vol. 4 of The Encyclopedia of Islam, New Edition, 997-998. Leiden, Brill: 1960
––––– “Mashrūbāt”. In Vol. 6 of The Encyclopedia of Islam, New Edition, 720-724. Leiden, Brill: 1960
Edit: Now that I think of it, I vaguely remember there being a line in the Arabian Nights where the protagonist is afraid of having his hand cut off for being drunk in public, but I'm not sure where I saw it. Will come back later.