Most questions I see asked about the death penalty seem to center on whether it is wrong because of the harm it does to the person who is executed.
What about the harm done to others by keeping a dangerous sociopath alive? Let's posit that we have a person who is so depraved that a prison sentence is no deterrence; and this person will gleefully cause pain, suffering, even death to prison guards and other inmates whenever he has a chance.
Is it reasonable for all these other people to have to be exposed to such danger?
Granted this scenario is an extreme case, that prison guards (let alone other prisoners) never anticipated such a danger to themselves when they first signed up for the job.
When people argue for capital punishment, one of the considerations they sometimes raise is deterrence. We can ask about general deterrence: does the death penalty tend to lower the murder rate? That's not your question. But we can also ask about specific deterrence: do we need the death penalty to keep particular, especially dangerous murderers from killing again? Someone could argue that death penalty statutes need provisions to deal with cases of the sort you've described: murderers who are likely to be a serious danger even if they're incarcerated. I'll confess that I find it hard to imagine a case where we had no other way of protecting guards and other inmates; far as I know, so-called super-max prisons already do that, though of course I could be wrong about how well they succeed.* If your question is whether there's a potentially legitimate question here, I'd say the answer is yes. But whether it will amount to an important part of a case for capital punishment, all things considered, is harder...