There are more considerations on prostitution that most everyone here is missing that need to be taken into account.
Prostitution is semi-legal in most places across the world. The local law enforcement definitely keeps them around and lets them do their thing. Of course, you try to keep street hookers out of the rich districts and they will sweep local rings every once in a while. But for the most part, they're ignored. There's a reason for that, that I'll get to in a sec. But as far as I know, this is the same practice in the US and UK, even though it's technically legal in the UK and technically illegal in the US.
The second consideration is drugs. Where there's prostitutes, there's drugs. You've got the street hookers, they've got the street drugs. You got the brothels, you got the party drugs. The two simply go hand-in-hand. Allowing prostitution while trying to keep drugs illegal is nigh impossible - the girls might use, the guys mostly use. This is one of the reasons the US has kept prostitution officially illegal, but informally semi-legal.
The third consideration is that prostitution is one of the few low-level crimes left. The police keep prostitutes around, especially in crime riddled neighbourhoods, because they're usually poor, young people who can't afford to spend time in jail. They can serve two uses. The first is that they know the neighbourhood they work in really well, and they hear a lot of things. The second is the probability that prostitutes may be open to the actual criminal underworld itself, and they're easy to push. If the police want to know the street rumor, or if they want to make a push on a local drug dealer, the first people they go to are prostitutes, because they know a lot.
Think about what would happen if you legalized prostitution in America. Gave them sanctioned brothels, gave them contracts and put them on a tight leash. The first thing you would see, is poor neighbourhoods starting to become slightly richer by the increased activity presented by these sanctioned sex hubs, at the cost of their reputation. The other thing you would see is a decrease in arrests. Not of prostitutes, mind you, but of murderers, drug dealers, thieves, etc; the people we want to catch, because the police don't have eyes and ears anymore.
Prostitution is a necessary evil, and we need to keep it evil for the greater good.
EDIT - I wouldn't mind a policy of having health checkups, though. Maybe informally. I'd imagine that if said prostitution is organized, that their handler/pimp/organizer already requires them to be healthy, but some will invariably be unhealthy, and keeping them healthy should be a goal. It's a side note that's independent of the question of legality, and I can't see a good reason to oppose health checkups.