Both sentences are grammatically correct but they can be interpreted differently.
I was born a girl:
The first sentence I was born a girl means that you came into this world a female. You were a female in the past but maybe you decided to change your female sexual characteristics at some point in your life and are now a male.
I am born a girl:
Whereas the sentence I am born a girl is stating a fact, that you are a girl. It can also be talking about right now, at this moment you are born and you are a girl.
If you want to tell somebody that you are a girl, you can say: I am a girl or I am female.