Summarize:

Since you use your non-dominant hand less often than your dominant one, its muscles are much weaker. This makes it harder to write well with your non-dominant hand. Increase this hand's strength by doing hand exercises that work these muscles and improve your dexterity.  Warm up and stretch your hands before working them out to avoid injuries and pulled muscles. Doing bicep curls with dumbbells strengthens the muscles in your wrists and forearms. Squeezing hand grips specifically targets your hand muscles. Something simple like squeezing a stress ball will also help strengthen your hand. Do this while you’re watching TV or on your daily commute. Many instruments require coordination between both hands to operate. Play one of these instruments to improve your overall dexterity and get used to using both hands together.  Guitar, for example, requires you to strum with one hand and fret the strings with the other. Since both hands need to work together, this activity improves your coordination. Other choice include bass guitar, piano, and drums. On many stringed instruments, you can flip them upside down and play them the opposite way with your other hand. When you get proficient enough, try this exercise to further strengthen your non-dominant hand. Any activity you do with your non-dominant hand strengthens it and gets your body used to using it. This will benefit your writing. Start doing more activities with your non-dominant hand to train yourself to use it in your daily life.  Try brushing your teeth and holding a fork with your non-dominant hand. Buttoning your shirt with your non-dominant hand is a good exercise that requires more precision. Don’t do activities that could be dangerous if they aren’t done correctly. For example, driving with your other hand isn’t safe until you’re very proficient with it.
Do exercises for your non-dominant hand. Learn an instrument that requires both hands to play. Use your non-dominant hand for more daily tasks.