Also a New Yorker, I grew up in The Bronx but spent four years of my young childhood out in CA with my mother. I took the subways at 12 by myself and with my friends. I've been flashed and chased, but the subway was the way to get around, so you just learned to be careful and things happened very very rarely anyway. After the incidents happened, I just laughed them off and Thanked God I didn't get hurt - chaulk it up to experience, right?
One thing a lot of people just don't understand: When you're a single parent and you're working day and night with no family, no friends to help you, your kids learn to grow up fast. I'm not talking about this woman in particular, but situations where kids shuttle themselves on the subway at a young age, or even get up, make breakfast, leave the house and lock the door and walk to school. When I grew up, just about everyone I knew was a latch-key kid (remember that term? I never hear it anymore).
Unfortunately, I have a neighbor (used to be a friend, but she just got too crazy for me) who will not even let her 15 year old daughter leave the fenced in (6 foot fence) backyard. She doesn't even know how to get home from the regular school bus stop so she begged the bus driver to drop her daughter off in front of her house. There's nothing wrong with the girl, our daughters grew up together and her faculties are all there. She even begs her mother for a little freedom, but her mother refuses. The mother is just extremely horrendously over-protective. She does the same thing with her other two kids.
Sorry, off-topic.
Would I let my 9 year old do it? No. Would I let my 12 or 13 year old do it? Yes - I did it and as a New Yorker, it's just normal.
Like the other NYer said, the City gives kids Bus/train passes. That's how the kids get to and from school.