I have a few suggestions that you can try.
1) Use wide masking tape to protect the screen. 1 inch or wider will work well. Plus use regular tasking tape not the blue easy release type.
2) Use a straight edge. A wide drywall knife would work well. Be careful of the corners as they might cut the screen.
3) Remove the screen then reinstall after painting. This usually involves removing some small wood trim. The trim can be painted before reinstalling the screen.
Removing the screen is the best way as all surfaces can be sealed from water, but is the most time consuming.