Cardboard boxes, and actually any wood products that are not treated (including tissue paper), are hard on fabric. Wood contains an acid that gradually eats fabric. This is one reason why many antique quilts that were stored for long periods of time in cedar chests have brown stains and holes in them.
To make cardboard boxes safe, you would have to either line them with muslin or spray them with a polyester film to prevent leaching of the acid.
I have stacking plastic tubs for my fabric. If you go that route, my advice is to skip the really big tubs because they get too heavy. I like the smaller tubs that are all see-through plastic. I tape a label to the inside of the tub (label facing outside), which keeps the labels from getting dirty and also allows me to replace the label easily. My very favorite tubs have latches on the box which make taking the lid on and off very easy, plus they stack really well. Unfortunately, Walmart isn't selling them any more. If you can afford it, it's really nice to buy all the tubs at the same time, as similar sized tubs from the store vary over time and often don't "play well" with existing tubs.