i've used the unvented gas heaters for about 20 years now: i keep a humidastat(misspelled) in a couple rooms: and have NEVER seen a big rise in humidity:
That could be for a lot of reasons, like the temperature you set, the amount of air leakage from outside, (including number of times the door is opened), climate, and the big one, the size of the area.
I would suspect, your home has enough air leakage, through tiny leaks like in doors, windows, fireplace, and from opening the door, that the moisture is not accumulating.
I have seen water literally condensing on the walls and ceiling in a 2 car garage, that was using an unvented catalytic heater.
We know gas, particularly propane, is loaded with moisture, and with an unvented unit, the moisture has no where else to go, but into the area your heating.
If you sealed this small barn with an unvented heater inside, it would be detrimental to the car.
He would be much better off not heating it at all.