Prostitution was. Prostitution is. Prostitution is going to exist till the end of the world.
There will be people interested in those services, and there will be those who want to provide them. Regardless whether I think it is a good thing or a bad, I think it should be legal.
Governments, police and anyone who needs a cause to fight against can continue to try to stop it. But it will be a lost war. They will never stop it from happening. They can try to oppress it, ignore it, bus as you cannot put a stopper at a hose without the pressure bursting a new way out for the water, there will be new places, new methods, new ways for it to continue.
Banning it will only achieve one thing: loose control of how, when, and under what circumstances it happens. The harder they fight against it, the further they'll drive it underground, and the harder it will be to achieve results. It is much preferable to be able to set the rules than letting things run wild and uncontrolled. Limiting the activity to brothels would make it easier to tax, safer for both the workers and customers.
So why not legalize it?
If it is a legal activity the laws can set “respectable” borders, can dictate where it can happen, can follow and monitor what can be done and how it goes. If it is legal, there can be tax income. If it is legal there will be no need of dealing-wheeling. It can be hidden away from sight without pushing it into dangerous neighborhoods. Yes, it is not quite the same as a massage, a wellness, or listening to a song, but they are all physical pleasures of varying degrees that can be bought. So they aren’t that different either. Make it legal, make money from it, and keep it between boundaries acceptable for both the workers and those who are against the trade.
If Romans could tax the public toilets why would prostitution be any different? The money does not smell.