Yes, you will need to cover it, or part of it. The easiest way is to have a piece that covers the entire screen top, then have cutouts that you can remove to add ventilation as needed.
Material:
A great material for this is the plastic corrugated "cardboard-like" material that inexpensive signs are made from. You can order it at sign shops or just ask them for scrap. It is inexpensive, cuts easily with shears, and is moisture-proof.
Your other issue:
To have great humidity while still having some ventilation, you need the cage walls to insulate well. Glass doesn't. That is much of your problem. How do you fix it? Tape styrofoam insulation to the outside of the glass walls. You can buy the 1/2" thick stuff in 2'x4' sheets and cut with shears or score deeply with a knife and break it.