I was taking the subway to school by myself at 12 in Toronto (maybe a slightly less intense city, to be fair). I'm not sure about NYC but in Toronto it's possible to teach a kid a set of strategies where they'll basically never get dangerously-lost.
Just find the nearest bus, and get on it. Stay on it until it arrives at a subway station. Ask a transit worker at the station if necessary. Get on another bus or subway that gets you a little closer in the right direction. It can take a while but you can follow this algorithm rather blindly and it will get you there. (The system was designed that way, I think.)
I never got lost. And if I had gotten lost, I wouldn't have been in any particular danger. At a certain age they'll have to figure out "oh, there's no bus, what do I do?" on their own. (At the time, I did not like this whole take the subway on my own thing - scared the heck out of me. I had to be coached with multiple trial runs.)