When should I close my screened bottom board?
If you don't live in a very cold region the bottom can be left open. I would suggest that if the hive is in the open that you provide some sort of wind brake. Some will stack hay or straw bales near hive and drive rebar down through the bales in the ground to keep them from falling. I use snow fence with black plastic fastened to it, to block the wind.
I leave mine open year round. In my area, during winter we'll go for weeks at a time of below freezing temperatures, and usually once or twice a winter a deep freeze of a week or two of sub-zero temps.