Or to quip: could slaves be patrons of baths?
Yes, they were.
The epigraphic testimony for slaves at baths as customers is quite direct, though sparse...
Thus, it is
difficult to determine how common and widespread the practice was.
Source: G. Fagan, 'Interpreting the Evidence: Did Slaves Bathe at the Baths?,' in D. E. Johnston and J. DeLaine (eds.), 'Roman Baths and Bathing' (1992)
The concensus among academics is that slaves could use, as customers rather than just attendants, public baths. For example, this from a blog from the University of Kent:
The rich, who had no time constraints, could choose to bathe at the optimum time and therefore temperature, but labourers or slaves — who were not in control of their time — had to settle for a tepid bathing experience once their work was done.
Also, Sandra R. Joshel, in Slavery in the Roman World states:
We know that slaves could visit the baths, if they had the fee...
For primary sources, Garrett Fagan cites an inscription:
An inscription from Puteoli prohibits the slaves who removed corpses in the town from using the baths before the first hour of night...
Source: Garrett G. Fagan, 'Bathing in Public in the Roman World' (2002)
In the same book, Fagan also cites the lex Metalli Vipascensis which
...stipulates that slaves and freedmen who were in the employ of the procurator in cgarge of the mine or who enjoyed other privileges could use the baths free of charge....the wording can be read to imply that slaves not in the employ of the imperial service would be charged, which in turn implies that slaves of any kind...had access to the baths as customers.
and an inscription in Coela, Thrace
...records the building of baths for "the people and the familia of our Caesar," that is, for the local inhabitants and the emperor's slaves who served his estates in the region...
Less reliable but still worth mentioning, there is also literary evidence from works of fiction. For example, the anonymous Latin comedy Querolus:
Even if we are tempted to go by day, it is at night that we go to the baths. We bathe with the slave-girls and boys...
Cited in Kyle Harper, 'Slavery in the Late Roman World, AD 275–425' (2011)
We know slaves were allowed within the baths, but the question is could they use the baths as customers. First of all, there doesn't seem to be any writings concerning regulations forbidding the use of the baths by slaves. We do know however that, even among Roman Citizens, access to the baths was regulated. The baths charged an entry fee.
They were public only in the sense of being open to all citizens who could pay the modest fee demanded for their use. Free baths there were none, except when some magistrate or public-spirited citizen or candidate for office arranged to relieve the people of the fees for a definite time by meeting the charges himself.
The Private Life of Romans, By Harold Whetstone Johnston
This would indicate that a slave would have to be of such a status that they may have had (or been given) the required fee to make use of the 'public' baths.
A good source which looks at the possibility of slaves as bath customers is the book
- Bathing in Public in the Roman World By Garrett G. Fagan, pgs 199-206.
Evidence presented seems scant, but does list a few occurrences of slaves using the baths. Again these instances that we have record of, were slaves of 'higher status'. There are also some good notations concerning special circumstances such as slaves in a mining region having certain hours set for their use of the local baths, indicating this use was allowed. From pg 204:
It is noteworthy that in all of these instances, the slaves in question either stand high in the slave hierarchy or enjoy close proximity to their owners; there is no evidence that menials would have had the same opportunities as these fortunates.
So it seems that if a slave had the status to have the entry fee, there was nothing forbidding his use of the baths on a customer level. But the scarcity of records supporting this would seem to indicated that it was not 'the norm'.