"Trail" is the main contributor to this particular problem. And, I am using a 1980's MTB, which has a lot of trail. In my case I thought something else is going on. The problem seems to be intensified because as I correct the steering (especially at slow speeds) the basket feels like it's swaying.

I can (and will) use a frame/fork combo with less trail but I didn't think this alone could be such a problem. With the weight shown in your photo (2 cases of glass bottles) my bike could not be ridden at all.