If someone has brutally murdered another person under circumstances where there is no doubt it was them, and there is no question as to the killing having been completely unnecessary and malicious, I believe removing them from society/the gene pool through execution makes perfect sense.
If someone's accused by the police of murdering their spouse, and the evidence is circumstantial... that's another matter entirely. When we have video or incredibly clear forensic evidence (victim's blood in their car, their semen at scene, murder weapon found in their possession, etc) I don't see any reason why we should keep them around to siphon taxpayer dollars.
We also risk the possibility that eventually subsequent generations may be stupid and release them. Destroying them prevents this.
In situations where guilt is not so clear, that person should have a life sentence and access to the internet, law books, whatever is needed for them to have a shot at proving they were wrongfully convicted. I would never support killing someone in an unclear situation even if the circumstantial evidence was strong. If there was not incontrovertible evidence like I mentioned above, particularly DNA... I would err on the side of caution and give a life sentence.
Maybe vengeance gets a bad rap. I used to be very anti-death penalty and that was a big part of why, I thought it was just blood lust/vengeance... but perhaps indulging in those instincts to a limited degree, with careful checks and balances, is important. Perhaps a society that turns it's back entirely on primal instincts like that just becomes a weak, frail society which invites conquest by other societies which still retain their spine.