In Basel itself only the Rhine comes to mind, but its not really know for carps.
Across the border in France the Grand Canal d'Alsac as well as the other canals and many small lakes are said to be good carp waters, and I believe that the one or the other catfish can be caught at the outlet of the hydro power stations along the Canal D'Alsac.
You can check with the fishing store in Birsfelden or with the ones in Germany (Weil a. Rhein and Loerrach) for details such as permits.
Especially the one in Weil a. Rhein has a nice range of good carping and catfish gear and, as far as I know, they also issue French permits (or check here: http://en.cartedepeche.fr/52-liste-des-cartes.htm
) The guys running the shop go to France regularly for carp fishing and they can be certainly an excellent source for information on good waters.