bern or basel

Bern is the capital and a very interesting medieval city. Einstein lived there and you could see his home and go to the Einstein museum. Basel is a gorgeous city on the Rhine with a lovely rathaus and town square. If it were me, I'd pick Basel and make a day trip to Bern.

I'm not a huge fan of either, but probably BERN - although 4 days is a long time for that area. I'd choose Lucerne.

It depends on your interests. Bern has wonderful museums (Paul Klee modern art in particular https://www.zpk.org/en) and many other sights that one might expect in even a buttoned down capital city.
https://www.myswitzerland.com/de-ch/reiseziele/bern/

Basel of course has its own sights:
https://www.myswitzerland.com/de-ch/reiseziele/basel/

Four nights of course means only tree days so you may want to plan a day trip elsewhere on the middle day.

Have fun.

To ChinaLake's point. If you are not already going to Luzern/lucerne, you should go there!

It would help to have more info. Are you exploring other parts of Switzerland? Will you be experiencing the Alps? What time of year? Where have you been and where are you going next? What are your interests?