With that said, I agree with Gaffer, I don't think the damsel in distress is as required as we have made her. It is a classic storying telling device, and definately not a bad one. I would say timeless. But as Tolkien showed us so well, it is not required for a great story. The hero simply needs a motive, a magic ring full of evil power was enough to get Frodo to travel to the ends of Middle-Earth through countless hardships without any promiss of winning the king's daughter's hand in marriage or winning the heart of his love interest and living happily every after. Of course Tolkien has been criticised for that, and the creative minds behind the recent film adaptions of the book found it fit to magnify Aragon's love story because of it.
But in Batman Begins, it was almost more of a childhood friendship thing than a true Romantic interest. If I remember correctly it seemed like she once had feelings for him but had kind of half way given up on him, and that he has always been too busy thinking of vengence for his parents to even consider her. But the relationship that was really focused on in the film was that of childhood friends. While he was off seeking revenge, she was still remaining true to the values of his family, and ultimately she is a major influence in opening his eyes to the greater good. I guess the movie could work without her, but why? I think the character carries more substance than the fast majority of hero love interests. She may fit the role of damsel in distress at moments in the film, but we don't have Batman fighting to win her heart. Certianly isn't a Mary Jane or Lois Lane sort of relationship. Maybe that will change in the second film, I really hope it doesn't though. I have always liked Batman being the kind of guy who is too concerned with doing his job to think about sex or relationships. Kind of adds to his sickness and obession to do good.