The official repair method is to buy a whole new rear half of the tub complete with bearings and seal for approx $180 (Sears does not sell the bearings and seals separately), but I have ordered those parts for less than $30. (I had read somewhere that the whole tub half has to be replaced because the bearings and seal are molded into the plastic tub, but this is not so: a steel sleeve is molded into the tub, and the bearings and seal are pressed into that sleeve.)
One I removed the tub and separated the halves it was easy to see why clothes had come out dirty: large patches of greasy brown stuff, a mixture of rust and grease, caused by failure of the seal, allowing water into the bearings and allowing the mixture of rust, water and grease to get out again and mingle with the clothes.
The rough patches on the inside of the tub turned out not to be the result of scuffing by the basket, as I thought, but simply hard residues, perhaps dating from before we installed a water softener.
The basket seems to be in fine shape, with no sign of the corrosion and failure of the spider that I have seen in photographs; the latter looked as though they were made of aluminum alloy, whereas ours seem to be pressed steel with some kind of coating -- no rust. We hardly ever used anything but High-Efficiency low-suds detergent; maybe this helped as well.
Here are instructions on pulling the thing apart:
and a video showing how to remove and replace the seal and bearings: