Earlier in sobriety, with closer acquaintances, sometimes I've said "No thanks. I'm good with club soda these days", or "no, those days are behind me, I'll stick with water". It doesn't out you as an alcoholic, but it also demonstrates that you've made a decision based on experience. It closes the topic of conversation, and I like that.
The phrase "I don't drink", makes us uneasy. It's too vague for some folks, especially younger ones. We also feel odd saying it, early on. Not sure why. Maybe it's sort of like the phrase "I don't have sex". Hmmmm....what does that mean? You've NEVER had sex? Or you just don't have sex NOW?
Of course, it doesn't matter what other people think. You don't owe them an explanation, and as others have said it's really not a big deal to most folks and it's all in our heads. BUT, for those of you who are early in and chewing on ideas, that one has worked for me. Obviously just my experience.