You pose an interesting question. When should a hero (or, more specifically a superhero, kill). I think the answer is never.
First, a historical perspective, in the Golden Age, Superheroes killed people all the time. In fact, if you look at the early run of Detective, Batman is pretty much a serial killer. See Batman’s Golden Age Kill Count However, at some point, there were public outcries that resulted in the creation of the Comics Code that comics publishers had to follow. And while heroes killing villains was not specifically listed, it would certainly have violated the spirit of the code. (“Scenes of excessive violence shall be prohibited.”) But, even before the Code, DC had a pretty firmly established a “no kill” in house rule about 2 years into Batman and Superman’s runs. As a result, while the Comics Code destroyed nearly every other Comic book genre (Archie also survived), it was the superheroes that thrived during the Silver Age. (In fact, a lot of the code was modeled after the code of standards used Archie and DC.)
There is also a practical reason why superheroes don’t kill. And that’s because comic book readers like familiarity and comic book companies like brand stability. Having an established rogues gallery helps with this and gives creators a tool box of established characters to pull from. The “no kill” rule makes sense. Comics are a world that relies on perceived continuity, where, like soap operas. sitcoms and Shakespearean comedies, ultimately the status quo must be maintained regardless of how upset the apple cart gets on the journey.
But, there is a deeper (In continuity) reason why heroes shouldn’t kill. Because if they did, they would be the villain. Heroes work, for the most part, inside the law. Their role is to police the city, jungle, world or sector. It’s not their place to be judge, jury and executioner. And ultimately, they put on a cape or a mask or a suit of armor or cape because they think they make a difference outside of the system but working with it. They think the justice system is flawed, but, at their heart superheroes have to believe the system works. Otherwise, they would be lawbreakers and not law enforcers. Ultimately, they believe the criminal justice system is ultimately fixable and desirable and work for it while they work outside of it. If a hero kill criminals instead of assisting the proper authorities in apprehending them, they are replacing the criminal justice system.
Sure, it’s easy to say that if Batman is justified in killing the Joker because it would save future lives. But, Ozymandias thought he was justified in killing millions of people because it saved billions. Where is the line? Similarly, if Superman Is justified to kill Lex Luthor, when? For what crime? Is it after his first failed land scheme in California, his attempt to steal LexCorp pensions or after he sides with Perpetua to end the multiverse? What in Superman’s background makes him the qualified person to make that call? In the first season of Arrow, there was a Huntress episode of Arrow where Ollie lectures Helena that he’s nothing like her because she’s a murderer. But, at the time, he was exactly like her. Ollie killed people who he were in his father’s book and had “failed his city” and Helena was killing people who had “failed her family”. Instead, it appeared that Ollie’s only problem was the people Helena was killing weren’t in his book of “city failures.” I think they eventually did a nice job of addressing that.
A hero has to stand for more. A hero is a person who takes the law into their own hands to make sure criminals face the legal repercussions of their actions. An antihero is a person who takes the law into their own hands to viciously murder criminals. Heroes have to believe in redemption (hell, most are trying redeem themselves). Killing removes any chance of that redemption. And while the concepts of the reformed villain is not as prevalent in DC as in Marvel, if heroes had simply killed the bad guy, we would have lost half the roster of the Avengers. (And no, I don’t think the Suicide Squad are not Superheroes 
). If you want to blame someone for the Joker’s future murders, it’s probably Gotham’s failed legal and penal system.
For some reason, the Heroes don’t kill rule hasn’t transferred over to movies, which is is a shame. And don’t get me started at how “I’m not going to kill you, I just don’t have to save you” is ridiculous. I could get easily get a murder 1 conviction out of that scenario. Batman creates the harm (By destroying the bridge), puts the victim in harms way (by destroying the controls), and then leaves the victim with no escape. I also think the only thing that should be snapping in a Superman movie is the dialogue.
Sorry that I babble. ‘‘Tis what I do. I guess I had stronger feelings about this topic than I thought. I started with a short answer.