Line them up on racks and make a top and side with shade cloth to keep them clean and still get air. You can buy shade cloth by the roll at a supply such as Home Depot in the garden area
Part of the curing process is evaporation. Moisture can't evaporate from soaps kept in a plastic box with the lid on. I don't think making holes in the lid would allow enough air circulation, either. Not sure about the spots on your soap, but I can't help but think they're related to being kept in a plastic box. Even a cardboard box would be better. Your babies need more air!