I say "No" and feel those who support it do not see the big picture. The issue as I see it is not prostitution itself, but all of the things that go along with it. Not disease or any of that (which could be reduced by legalization) but the various social problems.
For example, let's start with the basics: where do the prostitutes set up shop? Like any form of adult entertainment even the majority of people who want it, do not want it where they live. This is why zoning for even things like adult video/book stores and the like is already a touchy subject. It has less to do with morality about the subject matter (despite what holy rollers might want you to think) but a lot to do with the fact that you don't want your kids to be exposed to it, or your neighborhood exposed to the elements (perverts and wierdos) that it brings.
You set up the whorehouse and it's even worse, not you've got all of these guys (and girls) coming and going fromt he place at all hours of the night. Given the nature of the services provided your going to have problems over payment/quality of services and all kinds of other crap. Unlike illegal prostitution, this stuff is going to be resolved by the police, so of course your going to have the police going back and forth there pretty regularly, and of course you need to pay for the exta police officers to keep an eye on it.
Then on top of all of this it raises a lot of issues about the rights of both the buyer and the seller in the sex trade. Right now only ONE state pretty much allows this, and their various resolutions are not nessicarly going to slide with all of the states out there (it would not have been an issue up until this moment).
Then of course there is simply the fact that if you've got sex workers around, more publically exposed to society, there are the social problems of people dealing with them outside of a work enviroment, knowing what they do. It also then becomes a viable profession and of course that sounds fine until your son or daughter grows up deciding she wants to work in the sex industry as a viable career. Some people might think they could deal with that, but honestly... until your there you shouldn't comment. If I had kids I would not be happy about it. I'd prefer not to have "my mommy is a whore" on "what does your mother do" day at school and kids growing up saying "I want to grow up to be a hooker".
Especially when legalized understand that advertising and such is going to make this kind of thing look like the most glamourous, easy, and exciting thing in the world, and you've got to raise your kids in that enviroment.
Plus honestly just imagine the fun of this being a legal business, and then private eyes and employers and such setting up outside of all the whorehouses specifically to gather information. Walk by a whorehuse, your boss can have his guy use it as an excuse to fire you due to the "Company image" if need be. Private eyes can keep files of people in general just in case someone comes asking about an individual. Allegedly this kind of thing was a problem in Nevada (not sure how, and even if, they ever resolved it). Take that kind of thing and put it on a national stage and well... you've suddenly got major societal issues that despite what you might think are not going to be resolved easily.
Right now I think the US has struck the right balance by banning Prostitution throughout the nation, with ONE major exception: Nevada, which has basically become a national red light district. Problems have been resolved within that state to keep it functional as an exception (ie not for our society to function on as a whole), and from what I've run into Nevada state law and precedent is a giant mess because of it.
The way I see things Greyhound advertises that they can get you anywhere in the continental US for $69 or less. If your really that hard up, a weekend at a cheap motel in Vegas with a hooker is probably in most people's price range.
I'd rather have people do that for legal prostitution than worry about the problems in my back yard... and honestly, if I ever saw like my Kid's Teacher or Principal or something going to a whorehouse (legal or not) I'd have issues. Especially if they were like carrying a suitcase of sex toys or something... and honestly this kind of thing is going to be like sending up the bat signal for the majority of hard up pervs to start converging on those areas.