Can you use GameCube paddles on the Wii?

Yes. The original Wii has ports for four Gamecube controllers and two memory cards at the top of the system. Gamecube controllers can be used to play Gamecube games as well as certain games that support it. (Such as Super Smash Bros. Brawl and Mariokart Wii.)

As of Auguest 20th, Nintendo has announced a "family version" of the Wii that will not include Gamecube controller ports. Though a launch date has not been announced, those buying a Wii in the future will need to make sure they buy the correct model if they are looking play Gamecube games or use their controllers in general.