Some of you guys are seriously misunderstand why people consider prostitution a bad thing. You would have to fight thousands of years of precedent and the inherent complications of the sex trade - something which most people do not want to do, for very good reasons.
The most likely answer to the question would be that the US enacts a policy of looking the other way, and law enforcement doesn't pursue prostitution-related offenses, aside from sex slavery (in which the slavery aspect is being targetted, rather than the trade of sex itself). For the most part that is the policy in place today - escort services are common enough here, and there is an understanding with law enforcement and among society in general that prostitution isn't worth pursuing. Once in a while the participants are caught in sting operations, because the value of using a honey trap is too valuable for law enforcement and spy agencies not to use. A widespread utter revulsion at the idea of prostitution would deny that inroad to social control, which is why the sex trade will likely remain quasi-legal. Actually enforcing laws against prostitution in a consistent manner would be a great expense, and there aren't enough people with sufficient distaste for the practice. Even the ardent anti-prostitution activists are more concerned with using the law as a tool to punish people they consider deviants, and know they don't have the resources or political will to reform society and the law in a way that would produce meaningful results.
In order to have (even more) acceptance of prostitution as a legitimate trade, you'd basically have to undo the influence of Christianity - and the rise of Christianity is a direct response to the moral decay the Romans saw around them. People embraced the new religion specifically because it condemned the sexual attitudes prevalent at the time, because (for various reasons) they saw what was happening to the world around them, and concluded that their society had failed.
I would argue that if someone wanted to abolish prostitution, they wouldn't use legal punishments at all; the best way would be for all men to simply stop paying and refuse to hand over a single cent towards the trade. For that to happen, would require vast changes to society and how men interact with it, but unlike a world where prostitution is socially accepted (which cannot avoid the inherent consequences of the sex trade), it is much more plausible that, in the distant future, prostitutes would simply be starved of clients, and in my view the world would be a much better place by doing so.