There's a grey area that depends upon circumstance; this isn't really a cut and dry question.
A deliberate, premeditated attempt to kill Joker, or any homicidal murderer, super criminal or otherwise, would move Batman from vigilante, to murderer. It would be pretty hard for the justice system to overlook this even if he was essentially doing a great public service.
Plus, if the Joker is legally insane, which is another grey area considering he frequently displays intent, premeditation as well as an acute awareness of the criminal acts he commits, then can't be held entirely liable for his actions by law, which IMO is a bit ridiculous.
I'll just say that if in the attempt to apprehend the Joker that he sustained life threatening injuries, or was in a life threatening situation, I personally wouldn't make more than a token effort to save his life.
Batman, on the other hand, would most likely risk his own life to do just that. But that's what makes him a hero and not a murderer or even a regular person who wouldn't place themselves in danger to save a known and convicted serial murderer.