Unplug the Controller
Since cleaning the GameCube controller requires water, keep electricity far away from the device until well after finishing. Don't plug the controller back into the system for at least another day.
Water and Toothbrush
Using a cleansing solution stronger than water may degrade the electronics within the console. Get a cup of lukewarm tapwater and an old toothbrush. Dip the toothbrush in the cup, but don't load the toothbrush with water. Be as gentle as possible once you start cleaning to avoid causing damage.
Brushing the Controller Clean
Scrub around the controller's buttons and joystick. Make sure to get the brush around the sides of the buttons and control stick, since these areas get sticky easily.
Drying the Controller
Take a paper towel and wipe off the controller gently. Strong paper towels without much fluff are best. Avoid leaving bits of towel behind. Let the controller air dry for at least a day--you can put it near a fan to speed the process, but avoid heaters.