Some of it depends on your coop. I leave my one window open in the summer plus I have a lot of other ventilation up high that is open year around. Some of that higher ventilation does not have any overhang or rain protection, but it and the window are both on the side that seldom gets rain or snow. On occasions rain or snow does blow in.
I have a dirt floor and I use wood shavings as litter. When it gets wet it doesn’t take that long for it to dry out, mainly due to good ventilation. I’m kind slow opening it up most mornings too so they tend to scratch it up pretty well while they are waiting on me. Sometimes there are benefits to being lazy.
I don’t know how yours is set up inside, how fast it will dry out if it gets wet, or how much harm some wet will do. It is generally unhealthy for the coop to stay wet for very long plus coops tend to really stink if they stay wet for very long. I’d try to keep it as dry as you can but slight moisture isn’t the problem. Wet is the real problem.
There might be some trial and error involved for you since each of us have unique situations, but you might want to consider awnings or shutters to help keep rain out. Permanent ventilation up high, especially under overhangs, is usually a very good thing. If you know which way the wind is going to be blowing you can close the window on that side, but sometime you don’t win that guessing game or have enough warning.