Berne or Basel?

Personally I'd do neither and go hit the Bernese Oberland or Valais regions. Switzerland is like Australia, Canada or Scotland; nice cities but the countryside is better. If you insist on the cities Berne would edge it for me, but not by a lot.

Both are fine, but very different. It depends on your interests. Have a look at http://www.bern.ch/weiche_en
http://www.basel.com/en/welcome.cfm?
http://www.museenbasel.ch/en/index.php

There are tons of other nice places in Swizterland, like:
Entirely walled medieval town of Murten
Roman theatre and museum of Avenches
Menhirs around Grandson
Gorges de l'Areuse and Creux du Van
Watch museums at La Chaux de Fonds and Le Locle
Underground mills of Le Locle
Musical box museums at L'Auberson, Ste-Croix and Seewen SO
Celtic lake dwelling museums at Neuchatel-Hauterive and Biel/Bienne
Caves of Vallorbe, Reclere, Beatus
Ballenberg Eco museum and nearby Aar gorges
Baroque churches/monasteries at Solothurn, St. Urban, Einsiedeln, St. Gallen
Subtropical gardens at Brissago Islands, Isola Bella (Italy), Isola Madre (I), Villa Carlotta (I)
Engadin Valley (St. Moritz - Scuol)
Bernina railway
Mustair Monastery
Swiss National Museum at Zurich with dependances at Schwyz, Prangins, Gandria, Wildegg
Aletsch Glacier (Riederalp)
Rhine falls and historic towns of Schaffhausen (with castle) and Stein am Rhein
Medieval Gruyeres with cheese diary and nerby chocolate factory (Cailler at Broc)
etc. etc.
May be you will have a look at http://www.myswitzerland.com/en/home.html

fully agree with #1

i also agree on what number 1 said - if you really "insist" on a city, go for berne.
but rather go for the mountains..... in the bernese oberland, in the valais, or graubünden.