Cortina would be best I think - but do trains go there? If you look at this page you'll see a little map in which you can see that the mountains are very close to the town. Also there is bus service to the passes and to the other towns and villages in the area (Gabriele provided a link with the schedule on the page). I would say you don not need to rent a car there.
Tiziana should be able to answer about Trento - but it is not in the Dolomites. To reach the Dolomite mountains you would probably have to rent a car. Though you might stay in Trento and climb the local mountains.