Exercise both of your arms using various training methods such as weight lifting. Exercises such as curls, shoulder raises and the hammer curl press develop the muscles throughout your arm. Many players do not use their opposite hand during training and games, which makes it weaker than their natural hand. In order to be effective with both hands, both of your hands must be able to meet the strength demands of basketball.
Practice dribbling exercises to get used to moving with the ball while using your opposite hand. Train yourself to do such tasks as traveling the complete distance of the court, weaving in and out of cones, and changing directions while dribbling with your opposite hand.
Practice passing the ball with your off hand. Situations in the game may force you to make quick passes while the ball is in your off hand. Use your teammates as recievers and practice passing them the ball with your off hand while moving and standing still.
Practice shooting with your off hand. Shoot the ball from all ranges and angles of the court to become accustomed to shooting with the off hand. Layup, free throw and sudden-stop-and-shoot drills are all necessary practice to simulate the shooting situations that occur in the game. Alternate between both your natural and opposite hand to try and recreate your shooting motion.
Continue training dribbling, passing and shooting with your opposite hand before trying in a real game setting. Once you become comfortable with the movements during practice, only then should you carry it over to a real game.