"Crazy" is not a valid diagnostic term, just a petty insult. "Insane" is a term that has a legal definition but not a clinical one. Is Bruce Wayne legally insane? Absolutely not. He is aware of the nature and quality of his actions; he knows that he is fighting criminals in Gotham City instead of, say, battling Nazis in WWII or fighting demons in Hell. He is aware that his actions are extralegal, as evidenced by the steps he employs to conceal his identity. He does not suffer from diminished mental capacity; he has a brilliant analytical mind and a degree of perceptiveness that exceeds most people's, enabling him to be a master detective. He would thus certainly be able to understand the proceedings in a court of law and participate meaningfully with his defense.
Now, approaching it clinically, is Bruce Wayne psychotic? Absolutely not. He does not hallucinate or suffer from paranoid delusions; the enemies he fights are real, and he knows who and where he is. He does not suffer catatonia, excess or purposeless motor behavior, extreme mental preoccupation, or dissociation; on the contrary, he retains solid control of himself both mentally and physically beyond what most individuals are capable of. Were he psychotic, he would be unable to function as a crimefighter or maintain a successful secret identity.
It could perhaps be validly said that Bruce Wayne has a personality disorder -- perhaps something in the narcissism spectrum, an exaggerated sense of his own importance and the belief that he must personally defeat crime rather than trusting in the system. But I doubt narcissism would work, because it's characterized by a lack of empathy for others, and Bruce's whole life is about empathy for the victims of crime. It might be more accurate to say that he has an emotional disorder, inability to let go of the grief of his parents' death and move on with his life; but he has parlayed this disorder into something that's actually very constructive. Plenty of people can manage their personality disorders and direct them in a functional way, and only the most petty and mean-spirited of people would call that "crazy."
People often say that dressing up in a cape and tights is a sign of psychological dysfunction, but while that might be true in our world, it doesn't work as an argument applied to Batman. After all, he lives in a world where lots of people wear capes, tights, and other flamboyant costumes as part of a crimefighting career. It's a normative practice in his reality, and it's no more crazy for a crimefighter in the DC Universe to wear an animal-themed costume and mask than it is for a crimefighter in our universe to carry a metal badge around.