Minor edge wear is best fixed as mentioned above by Matt. To avoid further wear, try adding some bumpers on the sides to cut down future wear. Severe edge wear requires the Top off for a permanent fix and the bass half edged underneath with the new outer/top edge glued to the supportive half-edge under it. Sometimes this can be done superficially from the outside but will take a bit more work to un-do and re-do when the bass finally comes apart. Be careful what glues you use. Strong Hide glue would be the best because it can be loosened and re-done later. Any other glues can damage the original wood around it and make future repairs more costly not to mention possible damage to the bass itself.
Here's some Ebony and Maple Bumpers as well as some synthetic ones.
On the edges, here are some pics of inner half edging as well as some outer edges in progress during restoration.
It looked like this when it was all finished;
The restoration work done on the basses shown above cost more than the average bass or two or three or four! On old valuable pedigree bass restorations, you let the Luthier do it right and pay what it costs. The Bass will thank you later as it has helped to keep and/or increase its value over time rather then decreasing it with less that a top professional repair.