Basel, Bern, Lucern or Zurich for 3 nights?

Luzern definitely has the edge for me, because it will let you see the Alps! You could go to Mt. Rigi for a day trip, for example.
Basel has good art museums if that's your thing. And Zürich is a very pleasant city but to me, it is one of those places where I'd happily live, but don't feel an urge to revisit.

Also, to get from Luzern to the lake Como area you could look into taking the Gotthard Panorama Express to Lugano (a v scenic boat-train combo), followed by local train to Como town or bus to Menaggio depending on where you stay.

Lucerne is in the most beautiful setting. Bern is also
impressive, and makes it possible to daytrip to Murten, a walled medieval town that looks much as it did 500 years ago and Fribourg, another historic city. Bern also is close to the Berner Oberland so it’s possible to
even daytrip to Murren or Lauterbrunnen if you wanted
to see some spectacular scenery.

Another vote for Lucerne. I was just in all 4 places—Basel, Lucerne, Zurich, and Bern. I was a little disappointed with Bern, but really enjoyed the other 3. But, if I had to pick just one, it would for sure be Lucerne. In fact, when I return to Switzerland this June, I will be returning to Lucerne for a few nights.

As balso mentioned, Mt. Rigi makes a great day trip. This is one I plan to repeat in June as I loved the boat/train combo. Check out hotel des alpes. Switzerland has fantastic public transportation, so that won’t be an issue no matter what city you select. And for only 3 nights, I would definitely select just one location.

The correct answer is Zurich. Just kidding, it's Luzern.

I am joining the choir to say “Luzern”

I've stayed in all but Basel, and my vote is BERN by a wide margin, followed by Zurich.

Bern's medieval centre is unique, in that the whole area, several square miles, is authentically medieval. Its never suffered a fire or war which destroyed much of other European cities. From the train station to the river is a mile of traffic free shops, cafes, grand buildings, churches and medieval row buildings. And the famous Zytglogge (think gigantic cuckoo clock). For things to do, our highlights were the Bear Pits (more humane now than they used to be), the Einstein Museum, and the Munster (cathedral). Bern is also the capital and the Federal Building has a nice plaza and view. Murren/Lauterbrunnen Valley, Biel, Murten and Basel would all be easy daytrips. We went to Murren and Schilthorn once as a daytrip, and loved it so much we went back for 4 days the next time.

Lucerne is pleasant if you're passing through, but we had trouble finding enough to to hold interest for a full day, much less 3.

A few years ago we spent one night in Zurich and did day trips to Bern and Luzern. On another trip we did a day trip to Basel. All were enjoyable but if I returned to spend 3 nights in one it would be Luzurn - great combination of scenic beauty and an historic old center. Second choice would be Bern.

Have been to Zurich and Bern and wasn’t wowed by either. I would choose Lucern.