It is his responsibility in that it is the responsibility that he has taken on for himself. But that responsibility does not extend to the taking of life in the pursuit of his mission. That is the code that the character operates under. It’s self-imposed, but as such it’s the most important thing there is.

Turning this around on you a little bit: apart from because he has his name on the logo of the comic book, if Batman did go around just killing his foes all the time, what would make him the hero?