We're thinking of renting an apartment for a week in either Arles or Avignon and using it as a home base for car trips around the region. Which location would Fodorites recommend?

Avignon is more centrally located for the places you're probably thinking of visiting.

depends for the pont du gard, Uzes, the Camargue, Nimes, Aiges-Mortes and the places in the Alpilles. for Arles, very provencal.
Avignon, more sophisticated and a place to visit the perche villages of Gordes, and the Luberon.

Both make equally good bases to explore Provence.

Avignon has an attractive medieval center with papal palace, a pedestrian shopping zone, the famous bridge ruin, lots of options for dining, bars etc. Avignon is quite large and includes some industrial areas. I would say, it depends on the exact location of the apartment - use Google Earth to identify the neighborhood. If it is in walking distance to the center, it might be a good option. If it is located in an outlying business district, avoid it.

Arles has more small-town feeling. The town is dominated by Roman ruins, souvenir shops, street cafés, nice squares and alleys. Arles has a certain coziness (some call it sleepy).

As always, a matter of preferences.

We based in Arles for around a week several years ago and loved it. We found it centrally located for what we wanted to see, much of which was right in Arles (Roman ruins, Van Gogh).