If you want to use "abstain", it is more normal to say "I am abstaining from alcohol." It would be odd (for example) to abstain from wine, but drink vodka or other kinds of alcohol.
I agree that "I don't drink." is the most common phrase in AE and BE. so say that if your friend is fluent in English. If your friend is not fluent, then "I don't drink alchoholic." is clearer.
I've found when I "go out for a drink" with club members after our meeting (1 or 2 drinks and talk), no-one notices that some of us don't get an alcoholic drink. But it might be more noticeable in a place where everyone drinks wine. And if people are "going out drinking" (many drinks, getting intoxicated together) you don't go with them if you don't drink.
There seem to be many different customs about drinking alcohol with friends and co-workers, and I'm sure the customs are different in each country. They may be different in each city.