If there is adequate ventilation in the bathroom, most of the steam/moisture will go there. If there isn't, a door won't save you much, since as soon as you open it all the built-up steam is going to escape in the room anyway. Air conditioning generally takes care of it if it does happen though.

The extra humidity is bound to add to mold issues too. It's not a huge issue when it's largely contained to the bathroom where you can wipe stuff down, but mold in mattresses, upholstered furniture, curtains, and carpet make filling the entire hotel room with steam every day (if not multiple times a day) a very bad idea.

Open the window or run the aircon?

Good idea, I'll make sure the previous guests all do that.

It's been a very long time since I've stayed in a hotel room with a window that actually opened.

That's a comfort issue. Comfort is important, but it's distinct from hygiene.

  > it's distinct from hygiene.

Mold

Not to mention that any bacteria thrives in more humid environments. They aren't so good at keeping moist. This is true for a lot of things, especially the smaller the thing is, including bugs. Higher humidity definitely makes good hygiene more difficult.

Why do you think bathrooms have fans? That'd be a lot of effort to deal with farts.

Open the window or run the aircon.

This is a hotel room, you would need the last hundred guests to have done that, not yourself

Considering this and your other comments I really think you need to think a bit deeper about your answers. I believe in you, just ask "and then what happens" and I'm positive you'll figure it out.

I've both opened windows and ran aircons. (Though I try to avoid doing both at the same time.)

Nothing bad happened.

  > Nothing bad happened.

Keep at it, you're almost there. You just forgot about one important variable: time

If done both for a long time over many years.

If the increased humidity promotes mold growth, then yes, it's a hygiene issue.

Yes. Though trapping humidity in the bathroom doesn't make it go away, and you have to open the door to get in and out of the bathroom, and that lets the humidity escape.

The hotel will typically have an extractor running in the bathroom, wired to the light switch.