Sometimes, you can get foggy pictures because the glass on your box gets some moisture on it. If your box is not 100% leak-free (and I mean air, too), this is just going to happen whether you cover your box or not. Most of the time, the sun or drier air will clear the foggy glass up pretty quickly, and your cam will be okay.
Too big of a leak will hurt your camera. For instance, I accidentally closed my cam with a stick between the front and back so it closed with a huge leak for moisture to get in. The resulting moisture from some big rains killed the cam.
Now, I just cover the cam so that raindrops won't temporarily obstruct good shots. I've noticed that I often get my best day shots of animals on rainy or snow days, when they wouldn't normally venture out on a sunny day except at night.
I get a Tupperware box from Target, cut the bottom in half, spray paint it brown or camo or whatever, and put the half over the top of the cam. This works most of the time. I'll attach a pic. later.
I sometimes like shots of animals from near ground level and affix the cam way low. A big rain storm will splatter mud from the ground onto the box glass, ruining your pics. In that case, I lay something on the ground under the cam to prevent splatter. A towel, a foot mat, an old scrap of carpet, whatever you have lying around.