The people who mattered to me, by and large, already knew I had a drinking problem and were happy to see/hear that I had finally quit. Everyone else, I reckoned it's really none of their business. There are lots of stories of bad outcomes when people announce their problems to everyone, or get overly personal with TMI - quitting drinking is huge for us, and sometime we want to stand on a podium and tell the world, but most people either don't care or don't want to be reminded that they themselves may have a drinking problem.
What does happen rarely, with people I don't know well, is that I'll say I don't drink (like when a group wants to hit the bar at a conference, and I say I don't drink but I'll tag along and have an iced tea), and they will ask why not. This has only happened a few times, like once every year or two, but it has happened. My response has been basically, I already drank my lifetime quota, I'm done with alcohol. That has always shut them down. If it's someone I have a stronger relationship with, or want to, I might go a bit farther, that alcohol become a problem and so I quit drinking entirely. I've never had reason to go into more detail than that, with people who don't already know the story.