I have many decks still in their original tuck boxes, but for many and especially the ones I use the most, I've purchased tarot bags for them. Then I save the box elsewhere, usually flattened/deconstructed.
Tarot boxes that have lift-off tops I often continue to use as they are. If I want to use a box-set tarot with any regularity I will definitely take it out of the larger outer box and into a tarot bag, as I find those a pain to be opening up all the time. I love the Lenormand and pocket tarot decks that come with tins!
Then I have a few square or rectangular baskets that I put on my bookshelves, intermingled with my books, and the decks are stacked in those. Easy to access, and tidy!