If the box is the size of the tarot deck, I put it in a bag with the cards inside the box inside the bag (I usually have to find a little bigger bags). This helps protect the fragile boxes. For others that came without boxes, they get their own bags too. For bigger boxes (kits), I save them in their box, but probably will eventually take them out of their box. I only do this for sturdy boxes, otherwise in their bag they will go. I am getting too many now and some of my decks and guidebooks, as well as many tarot book are in storage. I have bought many bags (mostly purchased online), but I make my own too (so I can make any size I want). I prefer to keep the guidebook with the deck, but have broken up sets (kits) and shelved many guidebooks. I match the bag to the deck, as I have to recognize which deck is in which bag. So far, I only have 20 something decks. Most of my lenormands (I have really a lot) are in Indian style zip bags (gotten from an import shop), Most of my oracles cards are left in their boxes (because they're usually sturdy).
If you have space, I would try to keep them in their box (speaking from experience - I got rid of boxes and now I am sorry I did) and on a shelf somewhere. It's neater and easier to find decks once you have a lot of them. If you end up with bags, try to keep the boxes if you have the room. You may want them later. Also the bags can be expensive or time-consuming to make. Elle (sacredspace) on youtube buys inexpensive bags from gift stores/wedding places (sorry, don't have specifics). She puts a charm on each one to help identify it. I thought that's a good idea if you go with bags. Bags are harder to store than boxes. My bags are very decorative, pretty, and easy on the eye. I bought each one with a particular deck in mind, which makes it easy/easier to identify and find a particular deck.
Some people put them in small tarot-sized boxes (you can also buy little boxes that look like books from craft stores - not a tight fit). I store most of my decks in large decorative boxes that I got from Michaels or TJ Maxx, but bags could be put in a drawer or on a shelf. Right now, I have the decks I use the most along with some other tools (my runes, pendulum, astrology books) in a travel bag -- the size of a plane carry-on. This way, if something were to happen (fire, earthquake, etc.) I could grab it. I keep it in a travel bag because I take it with me when I see clients and it keeps everything I use organized.
I don't know if this helped at all, but I wish you the best of luck. Let us know what you decide!