If it were me, I would go out to the hardware store and you can buy sheets of acrylic. Get those cut to size and bolt them on to the sides and rear, and finally the front on pegs so you can lift and release it for access. Maybe do the front in several panels for each shelf so you only need to remove the one segment. You have the peg holes right there in the beams.
At least then you don't end up with a monument to dust.
