My GF bought the large Rad basket for her Mini step-thru and I installed it. We never thought about this, but because the basket is attached to the front of the bike frame with the 4 screws, it means that the basket doesn't turn with the front wheel but points in the direction of the frame (so it "lags" the turns, if that makes sense). After riding with it for awhile, she had me remove it because she said that it was affecting her sense of balance when riding and turning.
I have the Lowe's Craftsman 50 gallon bin and just used it for the first time a couple of days ago on my non-step thru Mini (but not with her bike and the basket). I did eyeball her bike just now, and it seems to me that the basket would be an impediment when folding in half and putting it in the bin. Next time I do it, I'll keep an eye out and report back...