You certainly can close them if you wish but I don't think you need to. We've camped in snow before in hybrids and popups and did not have a problem. We did clear the heavy snow from the bunks periodically.
If you do decide to close it up beforehand, then I really like the idea of using a dehumidifier. Also, use blankets for any runoff from the already-wet canvas into the trailer.
DW, 3 Kids and our Goldens
2012 Shamrock 233S
2008 Toyota Sequoia 5.7L 4WD