I'm an expat, and I don't drink. I also don't play most of the expat games (going on exotic vacations, talking trash about the help, etc.). So far, it hasn't hurt my professional reputation at all, nor my social life.
Professionally, it may have actually enhanced it. The expat community is quite small, and people talk (and talk and talk). I've tried to foster an image of someone who lives their life with integrity. I'm very good at my work, and I'm dependable. I will socialize, but I don't lose control. I try to be pleasant and professional in and out of work. Ultimately, when networking, people have hired me because I am an expert at what I do, and I do things well. I've never been hired for a position because I can hold my liquor, or because I got hammered or wasted at happy hour.
Socially, there are expats who aren't into the drinking scene. Find groups and clubs that share your interests. For example, I have friends who like hiking or biking, and they do that each weekend as a group, or sightseeing, or cooking...you get the picture.