I would be very cautious. PolyPro makes "breathable" covers, apparently. But breathable fabrics (whether it's Gore-Tex clothing, or RV covers) only work when there is a sufficient difference in humidity that it is able to "push" the moisture to the other side. Otherwise, it just traps it underneath. In an unheated building, the temperature changes will cause condensation to develop on all surfaces and you will be left with a soggy, rotten mess. A cover would be advised if it is stored outside, but only if it is the proper fabric and it is attached specifically and correctly.
If you're worried about dust and bird droppings in a barn, then a thin old cotton bed sheet, just on the top, would keep that gunk off but not trap any moisture. Or a canopy, or a suspended tarp.
After a year and half of storage in an unheated garage (in the Northeast) I have discovered that this moisture is insidious. From the start, I have put renewable/rechargeable desiccants in the cabin, with success. However, I have just discovered that I should also have put a small bag in the enclosed battery section, as the exposed electrical connections are now developing some rust. The galley itself is not a concern as it's all stainless and I haven't noticed any problems there.
I have used an impervious cover on my motorcycle in the same garage, to keep dust off, but I keep a fan blowing underneath 24/7 to inhibit any build-up of condensation in all the crevices. However, I don't think that this is a practical approach for a trailer and cover.
I looked for a previous forum post on this very subject of covers (I know there is one) but couldn't find it. Maybe someone can pull it up.
Steve