Many basketball players, even some of the pros, favor their strong-side hand -- the hand you write with, eat with and generally feel more comfortable using -- when dribbling the ball. And if you're a supremely skilled dribbler with your right hand, you can make up for some of your one-sided ball handling. But until you're able to dribble fluidly with either hand, opponents will be able to guess which way you're headed. So hone that off-hand now and see the results in the form of defenders left in your dust.
Resist the temptation to rely on your right hand while practicing. Naturally, you tend to use your better hand much of the time. It's more comfortable, you're more effective and it doesn't feel awkward. Tell yourself heading into practice that you're going to use your left almost exclusively. Sure, this may result in a few turnovers. You may even dribble the ball of of your foot sometimes. But it will also lessen that instinctual tendency to go right all of the time, and you'll eventually feel confident dribbling on either side.
Get yourself a blindfold or another piece of cloth that can be used as such. Tie it around your head and in front of your eyes to that you can't see anything. Standing in one spot, dribble quickly and powerfully with your left hand only. Once you've mastered this, try dribbling up and down the court with your left while blindfolded. This will help develop your feel for the ball with your left hand because you won't be able to cheat by looking down at the ball.
Practice even when you're not practicing. Football players with fumbling problems sometimes carry the ball with them everywhere in an effort to ingrain in themselves the habit of holding onto it, and this is a similar tactic. If you have hardwood floors in your house, dribble the ball with your left hand while waiting for the microwave to ding. I you like to go for walks or jogs, bring a ball and dribble it with your left hand while you go. Such frequent repetitions will further develop your comfort with left-handed dribbling.
Enlist some help. Have a friend guard you in a game of one-on-one, with a stipulation: he must angle his body on defense toward your right hand, so as to cut off any potential dribble-drives you might make with your right hand. When a defender "over-plays" one side like this, it's usually because he knows his opponent has one dominant dribbling hand. In this case, that will be true, only he will be doing so in an effort to assist you in developing your left.