I am a professional Fine Artist (with 2 small children). Storing artwork is a life long challenge for anyone.
The top challenges are acid, sunlight, moisture, extreme temperature & pests.
Most children's art is NOT done on "acid free" paper, therefor will eventually deteriorate regardless what you do. But you can extend it's life considerably if it is stored properly.
If the work is flat, like drawings and paintings, and you are storing it stacked together, then the best thing to do is put GLASSINE paper between each piece. This paper is PH balanced and is designed for this purpose. It is inexpensive & easy to find. (Michael's Crafts, or any art supply store)
I think Plastic bins are fine.
For absolute favorites, scan them & keep them digitally or have them framed professionally. A professional framer creates an air-tight seal that protects the work from moisture. You can ask to be sure UV glass is used & you will have some protection against sunlight as well.
Avoid storing in attic or basement due to moisture & temp changes. If you have little choice, go with the least moist option. (probably the attic)
Avoid hanging work in or near direct sunlight, it will fade and accelerate deterioration.
Avoid storing in cardboard boxes. Silverfish & other pests will eat the work gone. Especially if there is glue on it.
Depending on the amount of work you are storing, you can also get inexpensive paperboard portfolios with the ties on the side (great for flat storage, like under a bed) this will keep the work flat, but do not store these in attic or basement (or garage). Again, use glassine to separate the work.
Good luck, I hope this was helpful!
P : )