Sorry there's no consensus here, but for the destinations you name there's no great public-transportation advantage to Avignon versus Nimes versus Arles.
You might want to draw a little rail diagram: a triangle with Avignon at the top, Arles at the bottom, and Nimes off to the left (west) side. It's about 20 - 25 minutes between any of them by rail.
Draw a line to the east from Arles to Marseilles. Cassis and Aix are north and south, respectively, off of Marseilles.
Avignon offers some advantages, such as the TGV line that also goes to Aix and Marseilles. This is a faster train. However, the Aix and Avignon TGV stations are on the outskirts of town, served by shuttle buses, so your actual trip times may not be much faster than the regular train from Arles.
(The TGV stops at the regular station in Marseilles, so the advantage traveling to Casis might be better.)
Some of your other destinations may be accessible by bus, in season (when are you going?). Les Baux is closest to Arles, and Avignon to the Luberon area (possibly having to transfer in Cavillon). I really don't know about those wineries.
I happen to think that Arles makes a great base. However, maybe you'd like to rethink your preference for mass transit on this trip. A car would extend your reach and solve a lot of logistical difficulties. It also gives you a lot more flexibility should things go south two bus trips away from home.
If you decide to rent a car, you can base yourself in an apartment or gite (with a kitchen and space to spread out) in or near any number of small towns--St. Remy gets mentioned here a lot.
Whatever you decide, have a blast!