With the removal of God from law and government institutions, I question how the death penalty can still be considered immoral. If a man is guilty of crimes so heinous that the death penalty is considered, why is there any reason to keep him alive and healthy in prison? More to the point, what disincentive is there to keep people honest? If the worst you can hope for is "life..." that's not exactly a deterrence. (By the way, decisions of guilty are decided in court; appeal is a part of the process but if there is no new evidence, the decision should stand. Endless appeal is a problem both in clogging the system and in actually punishing criminals.)
From a secular POV, there cannot still be legitimate objections to "cruel and unusual death" or "inhumane" administration. We don't hang, electrocute, or use painful injections anymore. In fact, the death of murderers is now less painless and stressful than that of their victims.
Yes, there were a number of different factors that were influential in the intellectual debate concerning the death penalty and its abolition. Certainly the key ones you would need to focus on would be Englightenment thought and also how religious views towards the death penalty changed and adapted. In particular with regards to this last point, the sovereignty of God and how it is his decision to take somebody's life and not ours as humans was a key factor.
Enlightenment thinkers, especially Cesare Beccaria, were indeed influential in shaping new thinking about the penal system in general, but especially the death penalty. Jeremy Bentham, who doesn't exactly fit under the Enlightenment umbrella, thought similarly, even designing prisons that would be more humane. The Second Great Awakening also played a role- a major reform movement for prisons and asylums derived from new thinking about the sanctity of human life and the possibility for perfecting people through institutions.
As litteacher has noted, Michigan was the first state to abolish capital punishment. The influence of Enlightenment thinkers is indeed often mentioned in accounts of this development. However, religious arguments were also made for and again abolition. The matter was debated for a number of years before abolition was finally accepted and enacted. Here's a pretty detailed account of the matter: