Store the boxes in a cabinet or on a dedicated shelf, away from humidity and direct light - that way the cardboard won't warp. Use cheap transparent plastic boxes (like the ones from Ikea) to organize loose pieces by game, and keep everything in the original boxes or stack it all together if it takes up less space. Get those small zip bags to store dice, coins, and small pieces separately, and label everything with a sticker or permanent marker so you're not hunting around for stuff. If the boxes are pretty beat up but the contents are fine, put it all in a larger storage box that'll look better on display.