Personally based off of his personality it should be a Scorpio, Obsessive, intelligent, dark, mysterious, women constantly attracted to him. I read in a couple of places it was early march, which makes him an Aries, which doesn't fit him one bit- except his marshall attitude.
I know Dick Grayson (Robin I) was november 11, in the golden age, I think Scorpio fits batman better than robin, but if Batman was a Taurus with a Scorpio moon that could possibly work.
With his expert tactics and utter focus, I would think he would have a personality like Erwin Rommel (German feild marshall who did not participate in any crimes and tried to kill hitler- also one of the best tacticians in ww2) But that could be my own preference since thats my birthday.
That all stands to reason only if you believe in the retarded idea that the day of someone's birth is connected to their personality, attitude, or events that happen to them.
If the question was. when is BATMAN'S birthday? Then I guess you could argue that it is either the day his parents died, the first day he donned his costume or, the first day he made the headline of the Gotham Gazette (I think that's the name of the newspaper).