Cakes are much easier to decorate and more delicious to eat when the frosting is moist. Exposure to air causes the moisture in the frosting to evaporate and harden. This is problematic whether you have a batch of frosting that you need to keep moist while decorating the cake, or if you've already decorated the cake and want to prevent the cake frosting from drying. The basic principle for keeping the frosting moist is to keep the air out -- at all stages of the cake-making process.
Bowl of Frosting
Wet a paper towel or linen dish towel and wring out the excess water until it's just damp.
Lay the damp paper or linen towel over the frosting bowl. The edges of the towel should fold over the edges of the bowl, so air cannot enter and moisture can't escape.
Remove the damp towel cover when you need to get frosting out of the bowl. Replace it immediately when finished. Dampen the towel as needed if it dries out while you're working.
Cover the damp towel and bowl with a tight-fitting lid or clear plastic wrap if you want to refrigerate the frosting overnight. Paper towels and linen towels are thin enough that you should have no trouble snapping a lid onto the storage bowl even with the towel in place.
Place the cake on a cake-storage plate with a tight-fitting lid to prevent moisture from escaping the frosting. You can also use a glass cake display plate or an upturned bowl, but it doesn't work as well as a plastic container with an airtight seal.
Wrap the frosted cake with plastic wrap if you don't have a storage container large enough to fit the cake. The frosting will stick to the plastic. You must pull it back carefully, but it will keep the frosting moist.
Vent the storage container or plastic wrap before storing the cake in the fridge. The refrigerator can cause condensation to form on the frosting, which makes the frosting wet rather than simply moist. Pop the seal on one side of a cake storage container or leave a small corner of a plastic-wrapped cake loose.