They've always just asked me for my room number. I would assume that if you showed up with children, they would pull you aside and explain that the restaurant is adults-only. If there was a question as to how old a particular guest was, they could tell by looking at their room key. The last two digits of the year of birth are printed on the bottom of the card. As ducklite said, all of that info is also in the computer, so they'd be able to find out, if they wanted to.
I don't know this for a fact, but I would venture to guess that if one of the members of your party was an older-looking 17-year old that they wouldn't ask any questions. I have no idea how younger kids came to be eating there.