Airtight storage is key to keeping your donuts fresh, but the exact method and length of time you store them are determined by the kind of donuts you are saving.
To keep yeast donuts fresh for one or two days, store them in an airtight container or plastic sandwich bag without any excess air away from direct sunlight or heat.
Filled donuts are a different beast. Dairy-based creme fillings will spoil if left at room temperature, so they need to be refrigerated in an airtight container.
Jelly-filled donuts are equally delicate and will grow mold if left out, so store them in the refrigerator to keep them fresh and safe to eat for three to four days.