My small vacation apartment has no cooker hood and even the simplest kitchen task, boiling water, creates condensation on windows. I try to keep a window open whenever possible. At what point can high room humidity cause hardware damage?
There are several forms of damage high humidity can cause. Condensation on metallic parts may cause corrosion and combining condensation with the dust you get in any space occupied by people can clog up vents and overlay components preventing sufficient cooling.
However, you might find that you don't in fact have a humidity problem just a condensation problem which is unlikely to really impact electronics that much. You might get a humidity sensor to check but it is generally quite hard to reach levels of humidity that will actually be damaging as that is likely to need >80% for extended periods. That would be very unhealthy and will cause far more damage to you than the electronics. Humidity should be kept at around 40-60% for occupied areas.
Condensation on the other hand simply happens when the moisture in the air touches a surface with a temperature below the dew point. This can still be unhealthy as it breeds molds which can be quite dangerous to health. But this is unlikely to cause problems to electronics left in situ. You might get some problems with electronics that you bring in from outdoors and it may be wise to let them aclimatise for a while before using indoors.
Really, you should get someone to fit an extractor fan with as large a capacity as you can so that you are getting humid air out as quickly as possible.
THe short answer is "Yes", but that really depends on just how much humidity there really is. Certainly if it is somehow able to condense in your computer it can cause damage.
Most general-purpose computing hardware is relatively well-protected against environmental humidity. When operating, device temperatures will normally be somewhat above ambient, which reduces the risk of condensation.
The items you'll want to be most concerned about are tape drives and other magnetic media, especially if the tapes are stored somewhere cool.
Also, allow equipment time to warm up if it has been moved from a colder environment into the damp/humid one.