Firstly, it must be noted that there were drainage facilities for baths, so the water did get regularly replaced.

Secondly, it is interesting to note where said water came from. You see, the water going into the baths was, perhaps surprisingly to our ears, not supposed to come from the cleaner sources. Indeed, it came from the worst. Frontinus' de Aquaeductu ('On the Aqueducts') is the key source for this, and he lists the water sources of the various aqueducts. Spring water, such as that which supplied the Aqua Marcia, was the cleanest, whilst river water was worse (particularly that of the Aqua Anio Novus upon Frontinus' initial appointment as aqueducts commissioner), and lake water varied in quality.^a Later, he notes that the clean spring water was mostly reserved for drinking, and notably comments in chapter 91 that 'We have found even Marcia, so charming in its brilliancy and coldness, serving baths, fullers, and even purposes too vile to mention' after the dirtier river water of the Anio Novus began being used to fill up shortfalls in the supply. Chapter 94 includes a piece of legislation introduced by Frontinus, saying that 'lapsed' water – that is, water that fell out of the troughs – was 'not granted for any other use than for baths or fulling establishments,' clearly showing that the standards for bath water were not at all high. Obviously, this could not serve as the only water source, and so the dirtier Aqua Anio Vetus was also to be 'used for watering the gardens, and for the meaner uses of the City itself.' Again, that is not to say that only bad water was used. Agrippa's Baths were supplied by the spring-fed Aqua Virgo, which Frontinus notes had no receiving reservoirs or catch-basins for filtration, although whether this was due to its existing cleanliness or its usage is unclear.

So now we know that the baths didn't necessarily have the best quality of water. But how was safety ensured? Well, we know that when bathing, one went between a series of progressively hot baths starting with the warm tepidarium and then the hot caldarium, potentially followed with the sauna-like laconicum. The unheated frigidarium was a dip at the end. (Stambaugh 1988, p.203) Hence, most of one's exposure to water would have been heated and thereby to an extent sterilised. Clearly, as noted by Frontinus' complaints about the sources of drinking water, there was a knowledge that unclean water caused illness, so the use of it in baths must have been seen as an acceptable risk. Perhaps what was not appreciated was the possibility of infection arising from inter-personal contact. (Robinson 1922, p.113)

I hope that that has helped.

^a The water of the Aqua Alsietina, drawn from the Alsietinian Lake, is described as 'in fact positively unwholesome' by Frontinus, but during his tenure the Anio Novus was extended to draw from the lake north of Nero's Sublacensian Villa, the water of which 'bids fair to equal Marcia in all points'.

Follow up question: what did they do with the water after it was used? Was it used for irrigation, for example?