One argument I've heard for not having a death penalty (in fact, it was for not having equal punishments for sex crimes and murder in Australia, but given the US has death penalties for murder the distinction is not relevant at present):
If we awarded the death penalty for rape, then a person who commits rape has no incentive not to kill the victim; if they don't get an extra punishment, why not do so? Surely most people would rather have been raped and survive than be killed, so we should try and maintain incentives for rapists to leave their victims alive.
I can think of a few reasons why this argument isn't the greatest, though.
1) Most rapes are not random stranger attacks and are committed by people the victim knows, often in a context where the perpetrator can exert power over the victim to make sure they don't tell anyone. Consequently, many rapes go entirely undiscovered and unreported - noone but the perpetrator and victim even know that the crime has taken place. With a murder, while you're removing some evidence (i.e. the witnesses testimony) and improving your chances of being identified etc, if it wasn't a random stranger attack then you're also vastly increasing the chance of the crime being discovered because the person goes missing.
2) Many sex crime offences are defended on the basis of whether or not there was consent (and in some places, whether or not you knew it). This is generally a much easier fight to win as it is usually a he-said-she-said argument of credibility, and the system favours the perpetrator by limiting the extent that their sexual history can be brought in as evidence and not limiting the adducibility evidence of the victim's sexual history (i.e. if the defendant can make the victim look promiscuous, juries are often willing to believe "she was asking for it/she said yes").
3) It is very easy to argue that the level of punishment for serious crimes is either not contemplated by the perpetrator at all (crime of passion, e.g.) or if it is, is weighted very low because the probability of being caught, brought to trial, and convicted is perceived as being very low. That is, people don't commit these crimes because the punishment is not heavy enough, but because the perpetrator doesn't believe they will be punished for it at all. If that's the case, it doesn't matter what the difference between sex crimes and capital crimes is in terms of punishment, because the perpetrator doesn't consider it relevant in the first place.