Hold your strong hand behind your back and force yourself to dribble up and down the court using the weak hand. As you grow accustomed to the "handle," increase the dribbling speed. As with any dribbling drill, use your fingertips on the ball and do not look down as you dribble.
Practice driving to the basket from the weak hand side, and finishing with a layup or floater. Every layup drill covers both sides of the basket, but some players always shoot with the strong hand. Practice using the weak hand until you are confident that you can make a driving layup with the off hand.
Make an effort to use the weak hand for everyday tasks, like holding the phone or opening a door. This will help you get used to using the weak hand, so using it in a game is not an unnatural action.