I agree with egroen that nothing can be done to eliminate prostitution. Furthermore, I don't regret that fact; I have no interest in eliminating prostitution.
Child prostitution, and all forms of involuntary servitude, definitely should be eliminated. Decriminalizing prostitution seems like an obvious first step to me. Why should the authorities waste their time arresting prostitutes, who do, like it or not, perform a valuable service and who are more likely to be victims than victimizers? Decriminalizing prostitution would provide for greater transparency, and might make it easier to gather information about pimps and others who force women and children into prostitution or make it difficult for them to walk away when they want to. At the very least, it would free up personnel and resources for the task.