Kids can handle more than we all give them credit for. I was a kid not too long ago, so I know. Heh. My dad rented Terminator 2 when it came out on VHS back in 1992. I was 7 years old, and we stayed up until 1 AM (the latest I had ever been awake at this point in my life) while my mom and sister went to bed. Once and awhile, I'd watch Tales from the Crypt with him, too.
A movie like Training Day, or one of the SAW films, yeah, get your kid away from it.
Otherwise... who cares? You know? We're so worried about "protecting" our youth, we never stopped to realize that when they grow up and find out what the world is really like, they're going to be SOL. Prime example, have a friend who home schools his kid. Not to knock home schooling, but in this case, the poor kid has lived in a bubble. He's 12 or 13, and thinks "the S-word" is STUPID. He can't even fathom a word like SHIT existing. When he goes off to college, it's going to be like setting a toddler down in the middle of time square in NYC!
It's Batman, and it is rated PG-13 by the super-prudes of the MPAA. It's tone may be adult, but it's not as if it's racing with adult content or something. It's simply a more mature film that requires more intellect.