We're on the west coast of BC where it rains almost non-stop for 6 months over the winter. Have a cover on our TT in the off-season and crack a couple of roof vents and windows. Have never had mold. We use damp-rid, as do most. We have a hygrometer in the TT and humidity is a bit higher than I'd like but doing better than Damp-rid means a dehumidifier which uses power and needs emptying periodically (or draining directly outside).
The colder the outdoor temp. is, the less moisture the air will hold. Cracking a vent and/window shouldn't cause a problem. I would avoid ever heating the interior. Condensation of surfaces *might* occur, I think, if there is enough temp. change in the day like say when the sun comes out for a while then cools down quickly but if inside a barn might be unlikely. For those that camp in the winter, sealing up the interior to conserve heat is a bad idea.
The infamous "leaky condo crisis" of the 80s & 90s was caused by inadequate ventilation and leaking building envelopes (windows, siding, etc.). For many years, there were condos & other buildings all over draped in humongous tarps while they ripped off the exteriors, drywall & insulation down to bare structure and re-did it all. The building code in Canada was changed to require continuous ventilation, usually a bathroom fan hard-wired to stay on.
As mentioned, mold can be there and unseen. Take a food source like luan and just add water and voila... Some types are toxic. Can be difficult to eradicate sometimes, depending on type. Bleach will work in some cases, but not all. In the worst of cases, some homes can't be salvaged. If it's bad enough, the only way to eradicate it could involve pulling off the luan walls & ceilings.
I would get a hydrometer and see what the actual RH is. Maybe talk to a mold professional. Also check with your insurance co., if insured over the winter to see if covered.
