It largely depends what you do with your watch between wearing it and putting it somewhere air-tight really. If it was hot and there is sweat still on the watch and you put it straight into an airtight safe without allowing it to dry first, then there could be an undesirable level of humidity in the safe - I wouldn't have thought it would be that bad though, unless there were a lot of watches all on leather or fabric straps (that will absorb sweat more). The combination of humidity and salinity can be an issue but I think it also can be overstressed.

My watches are in wooden watchboxes - and I do actually have a couple of sachets of silica gel in each. For peace of mind as much as anything. But it's more important to take care of the watch generally - avoid water as much as possible, wipe dry after wear if it's sweaty and let it rest in somewhere with airflow before it goes back into storage.

It's far more likely that a watch will be damaged through misuse or mishandling than a humidity problem in a safe.