Consider a painting rack to store framed artwork on, wrap paintings in plastic to keep moisture out. For unframed art you can lay them in an archival box with acid free tissue paper between pieces. The basement and attic can be easy and tempting places to store art, but with temperature changes and moisture potential this may cause more damage. If you need to stack the paintings place padding in-between each piece to protect them.
Collectify is a great piece of software to save your collection details in as well as your household inventory.
When your young children create dozens of pieces in a day, consider framing a few favorites and using the rest for gift wrap or gifts to grandma!