I have painted my polystyrene cases with white paint to make them cooler inside the case for my summer activities on bicycles and trains.
If you want to seal the fabric and make it harder for the water molecules to get out of the case, you can give the fabric a coat of acrylic sealer and then a coat of acrylic paint in a colour of your choice. This will also make it much easier to keep clean, most dirt will wipe off. If you enjoy crafty stuff you can even use products like the Jo Sonya range to decorate the case, to make it fit the decor of the house and easier to leave out where you can find the uke to play. If you do apply paint, look at the zips, you need to avoid glueing them shut with paint.