Start by spraying some compressed air into the joystick housing to get rid of any large particles inside.

Apply a small amount of contact cleaner or 90% or greater isopropyl alcohol to the joystick and move the joystick around, and then wait for it to dry.

Only use electrical contact cleaner or 90% or greater isopropyl alcohol. Other soaps and substances might be conductive and damage your controller.

Make sure to wait for the contact cleaner or isopropyl alcohol to dry before reassembling.