the main concern for me if I were to stay in Atrani would be that you are stuck with buses for transport, or a walk to the ferry at Amalfi to go anywhere else. I don't love the buses on the coast just for the time it takes and possibility of having to stand (or not fitting on it). The ferries are easier, but their schedules are even more infrequent than the buses. Salerno would give you more options for transport, and if you're JUST using this base as a chance to go to Paestum, and one of the two coasts, Salerno makes the most sense. I actually have never visited the Cliento coast, but I know that many Italians head south from Salerno to the beaches than go to the Amalfi Coast. I am sure it is pretty and charming, hopefully you have done your research for places to visit.
Trains to Paestum aren't VERY frequent, I think I recall last time they were spaced generally an hour apart (spaced from about 45 min-an hour or so) but then sometimes large gaps of 2 hours. I would pick your trains (and buy your round trips at the Salerno station) and then have backups.
Salerno has GREAT restaurants, my favorite being Cici Rinella for a more special evening, but for normal dining my favorite pizzeria is Trianon, Spunzillo and Pinocchio are other fun ones along the lungomare.
The only crazy crowds will be locals walking along the lungomare or corso, I wouldn't suspect you would see many tourists (or at least in proportion to the amount of Italians) along your walks.
6 nights is a lot though, did you have ANY set in stone places you want to go on the Amalfi Coast? I have spent 4 and 5 nights in Salerno the past two trips I was there, doing day trips out of town (Vietri, Paestum, Capri, Herculaneum, Amalfi, etc) and if I hadn't lived there before, I might have gotten bored with another night. There are only so many times you can wander around the corso and Via Mercanti to shop or walk along the lungomare to take in the views. I would say Salerno is a good base for trips outside of town, but I would only be able to do one full day in town for sightseeing, with the Duomo, maybe a museum or two, the gardens, and possibly Castello Arechi above town. I normally combine my Paestum day (where I have lunch in Paestum) with a visit up the hill to Vietri for the ceramics. Herculaneum and Caserta are other options (though Caserta would be easier from Naples and the Herculaneum stop IS about a 20 min walk from the SLOW train from Salerno station) for daytrips if you would like.
It might behoove you to stay in Atrani for at least 2 nights (with a nice ferry ride to Amalfi and back to Salerno, its less than a km to Atrani from Amalfi, so if its safe you could walk) if you wanted to go to Ravello or Positano, though that would give you only one full day on the coast proper. You could very well do most of the day you leave on the coast and take a 4pm or later ferry to Salerno as well.