I have used a stripping basket years, but I have no idea why..it caused more problems than it solved..BUT..thats because I dont have one suitable for what I really need one for, ie, off the rocks or Pike fishing.
Basically, the one I have is too small for use when stripping lures. If I wear it on my waist it gets in the way so most of the time I wear it low down, almost on my thighs, and to the left.
Having said all that, for fishing with buzzers and nymphs I really should use it as for that its great.
For use with lures it would appear the favoured basket is a small laundry or shopping basket with a dozen of so holes drilled in the bottom and 100-150lb nylon line "spikes" inserted to stop the fly line catching.
Have a look at this informative video.