The first thing I do is press down on the lid without actually closing it, just resting my palm there, because that tells me whether it will sit flush or whether I'll be fighting it every time I want to tuck it away. A lid that rocks or needs lining up just right is a lid you'll stop using within a month, and then you've got an open box of loose stuff on a shelf, which defeats the whole point.
In an ordinary room, these work best as the quiet layer underneath whatever you actually look at, on a closet shelf, under a bed, stacked in the bottom of a hall closet where nobody needs to admire them. A box with a lid reads as finished in a way an open bin never does, so it sits well next to things like baskets of blankets or a bookshelf with actual books on display, it gives your eye somewhere to rest between the stuff you're showing and the stuff you're hiding. If you've got none of these yet, start with one plain size, something that fits a shelf depth you already have, and get a few of the same one rather than one of everything. Matching boxes stack cleanly and let you label by what's inside rather than by which box is which shape, and that's the difference between a closet that works and one where you're opening four lids to find batteries.
Where this goes wrong is putting a bulky lidded box somewhere you need quick, constant access, like a kids' toy corner in daily use, or a kitchen drawer substitute near the counter. Lids slow you down on purpose, that's their whole job, and a drawer you're opening ten times a day doesn't want that friction. They also crowd a small room fast if you size up without measuring first, a box that looked modest in the store can eat a third of a closet shelf once it's home, so measure the actual space before you buy rather than guessing from how it looked sitting on a showroom floor.
The one mistake I'd steer you away from is buying based on how the box looks sitting empty, because a lid that seems snug and sturdy in a shop can warp or gap once the box is actually full and stacked under something else. Load it the way you plan to use it, stack another box on top if that's the plan, and see if the lid still sits flat. If it does, you've got a box that will actually hold up on a shelf you forget about for a year, which is the whole point of the thing.