Let me first draw a distinction here between drinking and drunkenness. It's entirely internally consistent to draw that distinction, as drinking is generally regarded pretty neutrally in Biblical sources whereas drunkenness is regarded as a weakness and usually lumped into lists of another negative characteristics. Personally, I think that wine and alcohol in general made particular sense prior to modern sanitation, as alcoholic fermentation kills microbes and bacteria that might otherwise ravage water stores and those who drink from them.

So for awhile, even Latter-Day Saints used wine for the sacrament for the same reason that other denominations do - its symbolic resemblance to blood. Somewhere along the timeline there came the Word of Wisdom. We believe it to be a commandment; you're free to think of it as a change of policy. Now observing the Word of Wisdom as written is a condition of continued Church membership, as well as freedom from other chemical dependencies (they didn't exactly have Vicodin in the 1800s but it meets the spirit of the law). It isn't one of the central LDS beliefs by any means, but is one of the things that really peculiarizes Mormons in the eyes of others.