Keeper of Boxes...
I've got a different problem: I simply can't display all my figures at once. I rotate them, which keeps my office interesting. So naturally, I keep my boxes.
The boxes (some empty, some occupied) are in my garage, lined up sideways on an Ikea "Billy" bookcase. Because I'm in California, dampness isn't a real problem...but just in case, I keep a small de-humidifier running out there so the cardboard stays nice and dry. You can get dehumidifiers from boat supply houses, or the "Solutions" catalog. The one I have is called an "Air-Dryr." It only burns about 4 watts, makes no sound, and keeps mildew at bay. When I bought it I caught hell from my wife, but it's proven so useful so many times (ever get water in your car?) that she now makes no comment when I buy odd tool gizmos.
I also keep those little absorbent bags that come packaged with various things, and spread them around any stored paper or cardboard.