Not clean at all. The streets of Rome were already littered with scraps, human & animal excrements, dirty water, guts and more, so we can make an educated guess about how dirty the public Roman baths were.

Ancient Romans saw bathing as a social activity (as it was a place to meet and chill together), rather than a hygienic activity; meaning they would even eat food in baths in some cases. Often, citizens would go about their daily chores; sweat throughout the day and produce a lot of natural oils, before heading towards a WARM bath and scrubbing down. The baths had limited water circulation, no chlorination and often, a means of water getting in, but no means of it leaving. Surely, there would have been the usual sort of human effluent as well. No one used soap. Also, no one knows how often this bath water was changed...

So as we can imagine, the baths were hotbeds of germs and infections. Some ancient doctors (sorry, I forget who exactly; Celsus, I think) recommended not going to the baths with an open wound, as it could lead to gangrene, a gnarly life-threatening disease. Similarly, Estelle Lazer comments on diseases at Pompeii and Herculaneum, saying something along the lines of "the most common diseases were water-based", proving just how filthy Roman water was.

Also if I could piggy back on this question. How often did injuries occur at baths? Did they ever try to fix those issues that cause injuries?