Practice using your non-dominant hand throughout the day for simple tasks. Hold your silverware in that hand, comb your hair with that hand, unlock your car door with that hand and anything else you can think of.
Slow down and think about what you are doing when using your non-dominant hand. Focus on the movements the hand makes. Remember, the hand becomes stronger the more you use it.
Write using the non-dominant hand. Practice as often as you can. This helps the hand develop fine motor skills. At first, the writing will feel forced and look sloppy. Over time, you will notice improvement.
Stop yourself when you reach for something with your dominant hand and use your non-dominant hand instead. Eventually, doing things with the non-dominant hand becomes second nature.