I am sort of leaning to your husband and mother-in-law's side, although I do think that it is none of MIL's business and she shouldn't say a word to you about it. The thing is, letting your children take out their toys and strew them everywhere sends them a message that it is okay not to value their things, and it's okay not to clean up the messes they have made.
I'm sure that parts of games and toys are getting lost down there, if your children can take things apart, or open up a board game box, and then just leave them out. The next time they try to play with that toy, or that game, they won't be able to, because pieces are missing. My husband is a neat freak, so my kids always had to clean up their messes, with our help. But we taught them that if someone gave them a cool toy, or board game, or lego set, it was right to treat that toy with respect and put it away neatly so that nothing got lost, and the box it came in didn't get broken.
If you organize the basement so that everything (and I mean EVERYTHING) has a place, and it's easy to clean up, then have that be a nightly ritual. Your children will learn responsibility and respect.
(As for my putting the word 'everything' in caps, it really is the key to being neat and organized, in any room. If you clean and organize your kitchen, but you don't come up with a place for your keys, or your purse, or for newspapers to be recycled, or for batteries, or mail, or whatever, you will put those things down on the counter because there is nowhere else for them to go, and the clutter will come right back.)