Southern Italy city for a week in February?

Bari is certainly handy to go to these locations by train: Trani, Matera, Monopoli, Polignano a Mare and to Alberobello by bus. I’ve stayed in each of those cities - some twice, and I also stayed in Bari twice. The train to Matera is a different company than the one to the other locations.

I think because of the ease of public transportation, Salerno--pop. 110K and on the sea--is a great choice. We've based there twice, and it's the perfect jumping off spot for anyplace on the Amalfi Coast from Vietri sul Mare to Capri, then north to Naples, or south to Paestum. Great, reasonably-priced restaurants and lodging. Anywhere between 1 hr 40 mins to 3 hours by train depending on Freccia or regional train.

But much on the Amalfi Coast will be shut tight in February and if it rains there's not much to do.

Bari is a great choice; easy bus one-hour ride to Matera, among my favorite cities in Italy. Probably train to Lecce, or bus if not a train.

Naples would be excellent, better than Salerno without a car.

Another vote for Bari. Lots of places to visit nearby using public transport and plenty to do in the city itself.
https://slowtravelitalyspain.blogspot.com/2019/11/puglia-bari-in-april-trip-report.html
The Pinacoteca Corrado Giaquinto is worth a visit too

Bari is indeed a good choice, and I would also suggest Naples. Lots of possible day trips from both cities.

In February definitely use a regional or at least a provincial capital as a base.
The largest regional capitals in the south are (in order of decreasing size): Naples in Campania, Palermo in Sicily, Bari in Puglia. Provincial capitals that have enough to offer, to be considered are Lecce in Puglia, Salerno in Campania, Siracusa in Sicily, and also Matera in Basilicata.

Matera must be glorious In the winter!!

We were last winter in Termoli, went there from Rome. It's not south Italy, but it's really nice during this time. We did one-day trips to the Tremiti islands, and mostly were walking in the center. There is this blog from ferryhopper we got the inspiration from https://www.ferryhopper.com/en/blog/featured/best-winter-seaside-villages-italy and maybe you can see if there is anywhere else you might want to go. Enjoy your trip : )