Avon Snarksdale, after watching the animated adaptation of Under The Red Hood, can't understand why Batman won't kill The Joker:
But it also showcases a real problem for the Batman mythology. Later in the movie, a character gives an angry speech in which he asks why Batman hasn’t just killed the Joker, considering all the atrocities he’s committed. And that guy’s right. The Joker’s caught mad bodies, has beaten the second Robin to death, and crippled Barbara Gordon, the first Batgirl, by breaking into her house and shooting her in the back. The Dark Knight’s response is thoroughly unconvincing: “Because it would be too easy!”
Motherfucker, what? So thousands of Gothamites have to die — all those guards at Arkham Asylum, all those random civilians blown up or gassed or gunned down — because Batman, the only person who can ever seem to get close enough to The Joker to take him out, doesn’t want to violate his moral code? At what point is the blood of all those dead people on Batman’s gloved hands? We’ve all heard hypotheticals put forth by death penalty proponents in which the only way to save countless lives is to execute someone too dangerous to even be imprisoned. Well, the Joker is that hypothetical.
Obviously, there are larger plot reasons why Batman's greatest enemy can't be written out of existence, but Batman's explanation in the movie for why he can't kill The Joker rings false because it's so premised on vanity -- Batman is willing to allow The Joker to continue to pile bodies in the streets merely because he's afraid of losing himself to murderous rage. There are, obviously, civil-liberties implications to Batman's decision, but I've written about those before, so I won't revisit them here.
The right answer is that Batman is already an outlaw, but his extralegal behavior is premised on preserving and strengthening what legitimate authority exists in Gotham so that a better and more responsible society can ultimately be built. Batman exists because of the extraordinary circumstances in which Gotham finds itself -- a city so tainted by corruption that the local government is incapable of acting in the most rudimentary public interest.
So it isn't actually Batman's role to kill his enemies -- even those as crazed and sadistic as The Joker. That would undermine the larger project, the restoration of Gotham. Batman's endgame isn't a Gotham without crime -- it's a Gotham where the problems can be handled by public institutions rather than costumed vigilantes, where Jim Gordon isn't an anomaly. That can't be built on vigilante murder, particularly since doing so would make Batman a target of the very institutions he's trying to save. The moment Batman decides to kill the joker is the moment Gordon decides he's going to stop pretending he has no idea what Bruce Wayne really does at night.
To the extent that The Joker is still alive, that's Gotham's failure. Batman has captured him time and time again, only for the state to choose the lenience of a stay in Arkham Asylum over trial and state-sanctioned execution. The Joker, after all, isn't really insane in the sense that he's not responsible for his actions -- it's not like he kills because he doesn't know right from wrong. He kills because he thinks it's funny.