I don't like the idea that Batman is insane. I think it's misunderstanding the character on a fundamental level. He's obsessed, yes, but he's supremely rational and focused in how he manages that obsession, using it to drive him on a relentless quest to protect innocent people, to try to ensure that others don't have to suffer from crime the way he did. I think Nolan's presentation came closest to the truth; he doesn't dress up as a bat because he has some sick fetish, he does it because it's a symbol -- a piece of carefully designed and managed psychological warfare to strike fear into the hearts of superstitious, cowardly criminals.
Heck, in the context of the comic-book world where costumed superheroes are commonplace, it makes no more sense to say Batman is insane for donning such a widely utilized form of attire than it does to say that a mercenary is insane for dressing in military camouflage or that a football player is insane for wearing a brightly colored helmet with an animal's face printed on the side. In such a universe, the cape and tights are an established custom that serve a specific purpose and represent a specific subculture that Bruce Wayne has chosen to identify with. He not only wants to terrify criminals, he wants to give hope and comfort to the innocent, to reassure them that they will be protected, and that's what the superhero attire symbolizes. But Bruce has customized the superhero gear into very functional battle armor, which is another sign of great sanity and pragmatism.