Free market economists will agree that, while there is a demand and a supply for transactions that involve sex, that the market will find the most efficient way of providing opportunities for those transactions.
The problem that we think certain things are very important that people don't really believe are that important. For example, we might want to reduce the spread of STDs, where most solicitors of prostitutes might not. The market might develop two different markets; one for high-end (STD free) and one for low-end (potentially infected) prostitutes. I do not know how this market will form or if this is even the form this market will take, but I do know that the free market will result in whatever markets are best for the trading of sex and money.
While we push our values into the equation (sex is bad and should not be performed, STDs are bad and should be limited, prostitution is bad and devalues women, etc.) we will ultimately need a government to enforce our values onto the rest of the population, who may not agree with our values. But you're not perfect, and most likely your values suck for everyone else who might try to apply them in their life. It would be better if we didn't try to attach some of our own values to this scenario, and instead allowed for individuals to decide what is important for them.
Ultimately, the free market will provide the most efficient means of transacting sex for money if allowed to, and many of the "problems" we would find with the free market is that it would allow individuals to act against our arbitrary value system.