People focus on the salient risks. OMG, someone might steal my passport!
Fair enough - they might. But truthfully, how high is the risk of this if you're staying in a decent hotel and it's somewhere not in plain sight, such as in a bag?
I submit that it's not very high. The only guy I know who ever personally got anything stolen was while staying in a dorm room in a backpackers, and it was stolen by the other guy in the room, not the maid. As it turns out, the backpacker stole his MP3 player that he'd fallen asleep while listening to, right from out of his ear! Talk about chutzpah. We'll file that as 'one more reason to avoid hippies in backpackers'.
But a low risk of theft is, on its own, no reason at all not to use a hotel safe.
On the other hand, if you're anything like me, do you know what the much bigger risk of you being separated from your passport is?
Leaving it in the damn hotel safe when you check out of the room because you forgot to get it out.
I've done that at least once, years ago, but thankfully I remembered when the taxi was only halfway to the airport.
It's not a salient risk, but it's much, much higher.