Where to stay in Provence-no car

Maps will tell you where the trains go. If you want to go where the trains don't go, you have 2 choices: rent a car or sign up for an organized van tour.
If you have a car, you can go everywhere--except note that the cities/big towns in Provence often present significant parking challenges that make taking trains to these towns a more pleasant option.

Nimes, Arles, or Avignon each have bus and train connections to a partly overlapping set of day trip destinations. Direct buses to Pont du Gard run from Nimes or Avignon, for instance, but not from Arles. However, I'd generalize that wineries, lavender fields (if in season) and hill towns are better reached by car or organized van tour. Rick's Provence & French Riviera book covers more destinations in this region (versus his France book).

Nimes is the shortest travel time from Barcelona, 3.5 hours by direct TGV. How to Look Up Train Schedules and Routes Online gives you the DB train schedule link and tips for using it. And then will you return to Barcelona or move on within France?

If you want to visit wineries, see lavender fields, and visit hill towns you will either need to rent a car or take a guided tour. If you don't mind guided tours, Avignon is a good base with lots of day tours you can take around Provence and it has good train/bus connections to other larger towns you may want to see. Personally I would rent a car and go wherever you want to go - if you do rent a car you can pick a nice smaller town or village to base in because you won't need the good train/bus connections.

I think without a car the best location for train connections is Arles.
http://www.beyond.fr/map/ter_paca.html
One rail line heads to Nimes+Montpellier
Another to Avignon+Orange
Another to Marseille+Aix-en-Provence
Buses to Les Baux+St Remy

We have comfortably travelled to Avignon, Arles and Vaison la Romaine all without a car. This year, we will be traveling by train to Isle-sur-la-Sorgue and Nimes on our way to Barcelona.

I think Arles would make a lot of sense for a train-based trip, only 18 minutes from Avignon on the train according to bahn.de, less than an hour to Nimes. Without a car you would have challenges visiting the Pont du Gard and Les Beaux, both of which were highlights for us on our two night stop on our 2003 trip. It was insanely too little time as we arrived in Arles around 9 p.m. our day of arrival. We had a rental car, looked around Arles in the morning for a couple of hours then on to those two stops. We left thinking it would have been nice to have more time in Arles. I could easily see on a return visit spending two weeks in Arles and the Riviera alone. Or, add the French Alps which we visited on a 2010 trip. Southeastern France has so much to offer, and Provence perhaps has the most tightly concentrated places or worthy places to visit that are well connected by train. I am sure you will make the most of your four days.

Arles, for sure.