It doesn’t matter if it’s a restaurant, a bar, or a club. If one plans on drinking in NYC one should always carry valid government identification (student ID doesn’t count). ID laws are very strict here. Some bar’s & nightclub’s own policies regarding ID’s can be even more strict yet. So set your ego aside when restaurant/bar staff asks you for ID, and don’t embarrass yourself with a joke about being flattered to be asked to show ID… It’s just the law. ID policies also change based on the type of venue, night of the week, and even the hour of day. ID policies may restrict you from ordering drinks while inside, or prevent you from even entering at all (even if you are 1,000 years old)! The more notorious nightlife neighborhoods are far more strict in enforcement of ID policies due to random visits from law enforcement being very common.
Passports are the only 100% guaranteed ID, but understandably you’re concerned about losing your passport. Well, foreign drivers licenses can work too, unless it looks old, expired, damaged, and therefore suspicious/possibly fake. It would be best to bring additional forms of ID to back up your foreign ID card. A copy of your passport (or other photo ID) would be helpful in this case, and even some credit cards also bearing your name. Student ID could potentially be helpful in this case too, but never try to pass student ID as your primary form of ID.