Most times, games are easier to store in their original boxes. If you still have the boxes, can you just use clear packing tape to fix them? That would be the easiest and least expensive method.
If you don't have the boxes, or they are beyond taping back together, use clear plastic totes so the kids can easily see the contents.
For smaller games and puzzles, use shoe-box sized clear plastic totes. These can be stacked for easy storage.
For larger games or to store multiple games together, just get the larger clear plastic totes. Stack the boards at the bottom of the tote, and then use zip-lock bags to store game pieces. Obviously, you'd separate pieces by game. Then, label the the bags so you have each game in its own bag(s). This would eliminate the need to paperclip bags to the boards (I don't think paperclips would stay on game boards very well).
So, when the kids want to play Monopoly, they grab the board and the bags labeled 'Monopoly' and they're set. When it's time to clean up, pieces go in their baggies, and with the board, it all goes back in the tote.
Organizing kid stuff is an on-going thing. Just when I think I've got it down, the stuff multiplies, somehow. Looking forward to what others have to say.
Good luck and just know there isn't one right way to do this. I've often been stuck in trying to find the "perfect" and "permanent" solution only to realize that just about any system is better than constant chaos and toy land mines all over the house.
J. F.