Euro, I'm fairly certain the actual legal answer to your question is
yes, as a matter of law it's completely OK, legal in NY State to go in to a place which sells alcoholic drinks, if you are under 21 and you don't actually drink alcohol.
Note that - of course - some/many establishments have a sign "21 and up only" or "25 and up only" or "tall redheaded 40 year old people only" or whatever, that has no connection at all to alcohol laws. To be clear it's perfectly legal for bars in NY to discriminate (so to speak) against under 21s.
Note however, putting aside the law, we need a current New Yorker on here to tell us how common it is that bars/nightclubs simply do not allow under 21s.
Note that if you're thinking "cool nightclubs" (aside - why go to NY? you're 70 years old?! :) it's not the 60s anymore) they do (although this can change rapidly) often allow only over-some-high-age because it's just too inconvenient for the bar staff to check everyone. (Indeed, conversely back the 90s a "trendy" thing was actually all-night nightclubs catering specifically to youngsters, like 13 etc, obviously with no alcohol at all sold.)
BTW not specifically an answer to your legal question but you can easily google up articles such as
A recent article from the local TV station