So I had to come up with a solution to at least contain the mess and make it easier to move it up the stairs when needed. Cue Stair baskets. I have also heard them referred to cr*p baskets, but we won't use due to little ears in our house. I found some great baskets at Homegoods and personalized them for each child.
Here is another solution I have seen from Ballard and some other catalogs. I am not sure how I feel about it. It would definitely take up less room on the stairs, but it would not hold anything. Mine are normally overflowing as it is!