Having broached the "Where do you..." topic in an earlier post, I am going to revisit this. The question raised itself, unlike the nagging Skyrim bedrooms debate, in re-reading a few early Tom Clancy thrillers. There seems to be an oddly frequent number of occasions when Soviet officers in Moscow visit the public baths together. A few different critical readings of this could be developed: as a comparison between Russian and American culture or as a deliberate literary device to render alien the Soviet enemy. For my money, this was something that just stuck in Clancy's mind; it is hardly the only cultural difference that is raised, nor is it the only mention of bathing practices - we are asked to consider the difficulties of bathing aboard a submarine.
Now, bathing is something that players in a tabletop RPG are probably going to do less than sleeping. The benefits of sleep and rest are obvious; bathing perhaps less so in character sheet mechanical terms. Insisting on regular bathing in an RPG might well be overdoing it in terms of fine detail; much as noting the lavatory visits of player characters might be a little too much information.
The one time, in fact that I have made use of bathing in-game was concerning a healing spring. There is a definite trade-off. You heal, but slowly - and you wouldn't want to take your chainmail or spellbook into the pool with you.
There are some further uses of in-game bathing and sanitation that I can see - beyond the possibility of killing rodents and/or Harry Lime in the sewers.
Introducing a new culture might make mention of bathhouses; a visit to the mighty metropolis of Urbs Aeneae (or whatever Pseudo-Roman civilization happens to be in your neck of the woods) might well point out the bathhouses on a journey into the city (not that the Romans were the only ones with public bathhouses). I quite like the notion of debate regarding the water supply: the magical lobby maintains water elementals to power the aqueducts against the wishes of the opposition, who regard magic as unreliable and wish to install a purely mechanical system.The question of the gender mix also comes into play, as does if different species bath together. It need not effect a player, but it is a quiet reminder of setting.
Equally, pointing out a lack of bathhouses and the presence of bathing places on the river, along with citizens drawing water from somewhere upstream is instructive: the players know there will be no sewers to kill rodents in!
This is all rather secular; cultural tone might also be well served by purification rituals conducted in places of worship. Characters must cleanse themselves before entering the temple - a time consuming process - or (as cribbed from this post over at Roles, Rules and Rolls) immerse themselves in water to be healed - which may not be terribly efficient in the field. Speaking of purification raises an interesting time-management aspect to a game: the Church will pay you to slay demons or retrieve black magic artifacts - but then enforce several days of purification rites in order to swab any taint from you. All very well if you are being paid by the day, but if not deeply frustrating and restrictive.
Aside from this, the bathhouse might well make for a good setting. Spinning one out into an entire megadungeon is a little much. but as a setting for intrigue or assassination attempts is certainly interesting and forces some restrictions on play, calling for improvisation. Turning full circle back to Russia, I recall this happening in one of Boris Akunin's Erast Fandorin novels. Grappling with lack of weapons or armour or spellbooks is a problem; slippery floors and crowded spaces likewise - and does your assailant have any identifying marks, or are you going to have to threaten every tall blond with a red and white towel until you find them?
This offers a chance for a certain lightness of tone - think of footchases and outraged patrons - visually if nothing else: white marble or terracotta replacing the ten foot dungeon corridor.