From what I understand:

STDs have always been around, but we have to keep in mind the context in which the people that have the disease perceive it?

The biggest problem that we have with Ancient Rome and Greece is the lack of knowledge they had to describe their situation! I mean they wouldn't have had an idea of what an STD was; at the most they understood that there was disease.

So the answer is yes, but there's also the second part of your question with the (Roman) bath houses.

It depended on what bath house you went to for how disease infested a bath house was, but I would venture to guess that the bath houses had a lot fewer diseases then what you may guess. These bath houses were huge and well funded. They we're cleaned fairly frequently, and the Romans themselves clean frequently. Bath houses in the time of the Ancient Romans we're really sophisticated, and if you ever get the chance, go to the Baths of Diocletian. You will truly realize how insane and large these baths could be, but on the other hand, some were obviously less financed by the Roman Government like the make shift baths that can be found as far as Briton for soldiers. It truly depends where you look for this answer.

TL;DR Yes, but they're hard to find in historical literature because Greeks/Romans didn't understand the concept of an STD/STI.