As always “what’s the best xxxx” is not a good Question. In almost all cases there isn’t a best but instead multiple good options to pick from, and that is very much the case here.

I am concerned about protecting the wood in a humid/damp environment.

You don’t have to be. People worry about this sort of thing a lot these days (I did too early on) but there’s no reason to — things were made from wood and used in damp, sometimes very damp, environments for centuries before modern finishes came along and they lasted well.

You can even leave the thing bare and it’ll do just fine. Although obviously most people would prefer the wood to look ‘finished’ and for it to be protected a bit from stains etc. the wood itself just sitting there won’t be harmed by steamy air once or twice every day and the occasional splash of water.

I see polyurethane varnish used as the default finish for a lot of things these days. However, I find that (where I live anyway) there’s a wider range of non-poly varnishes available than polyurethane varnishes.

Poly is nearly the default choice these days because most consumer-level varnishes are poly, including when it isn’t specified on the labelling. And it is the toughest of the common varnishes (by quite a margin) anyway, making it the superior choice if durability and waterproofing are desired.

Specifically with regard to protection from moisture, what are the advantages or disadvantages of polyurethane or other kinds of varnish?

I’m presuming here you’re asking largely about protection from water vapour/humidity rather than liquid water, and there isn’t a lot between varnishes at a typical coat thickness used today. But referring back to my earlier point you don’t need to worry about this particularly so it doesn’t matter much here.

Are there any differences between oil based and water based varnish?

Yes. Very broadly speaking the former are better and the latter not as good. But, waterbased finishes have improved greatly over the years and some are now much better than they used to be, equalling the performance of oil-based varnishes in direct comparative tests….. old timers take note!

In addition to the functional aspects of the finishes, waterbased finishes are mostly “water white” once dry and change the look of wood minimally as a result (on pine and other lighter woods, they can give a ‘cold’ cast to dark woods).

Are there any other kinds of finish that I should consider?

I don’t think so no. Totally a judgement call of course but there’s no reason you need to look beyond common consumer-level finishes for something like this. I’m not a fan of them but commercial “Danish oil” finishes would probably be fine for this, unless where it’s sited it will be frequently splashed with water in which case varnish would be a better bet.

Just a final bit on this since it has more general applications to finishing:

Ideally, I would go for something not too glossy – a silk or mid-sheen finish.

All gloss finishes can have their surface modified after they’ve hardened to a desired sheen level. This was commonly done on shellac in the past and it works just as well today on varnishes and other coatings. I’ve even done it to paint.

The idea is very simple, you’re just scratching the surface very finely, usually following the grain, and being careful at corners and along edges not to rub through. The coarser the scratches (this is relative, they’re still tiny so you won’t perceive them as scratches if you do it well) the more matt the finish.

For a satin or semi-gloss finish you can sometimes merely buff the surface with a cloth to knock sheen back one notch from fully gloss. On fully cured polyurethane varnish because it’s very tough you’d usually have to resort to abrasives, so a gentle rub over with 0000 steel wool or the finest grade of Scotch-Brite will do the trick.

For something more mid-sheen and approaching matt you can use 000 / 00 steel wool or a coarser grade of Scotch-Brite.