Death penalty has its pro and con but I think it has little effect on reducing crimes.
There are no pros to it in my opinion, and like you said it isn't a deterrent.
Though it does help reducing cost spent on prison.
But does it? It might save on costs of housing the prisoner in the long run, but you're forgetting all the extra costs there will inevitably be with all the process and appeals and lawyers and extra time a death penalty case will go on for and they go on for many years.
Is there alternatives to punishing crimes?
Yes. Just actually life in prison, and life actually meaning life not just 20-35 years or whatever it might be (it differs from country to country). You can argue that putting someone to death is them getting off without doing their time as well (though the death penalty is obviously the ultimate punishment) and they can still get to 'enjoy' life (though I'm not sure how enjoyable life would be in a maximum security prison). Unless you're like a truly insane mass murderer that is guilty beyond all doubt then I don't think the death penalty should be even considered.