Brining is simply a way to increase the amount of liquid in the cells of the meat, to make it juicer, and it certainly will work well for a turkey breast. But you don’t need to brine it endlessly – three hours or so should be plenty (a whole turkey is generally brined overnight in the refrigerator).
Beyond that, the key to keeping meat from drying out is to not overcook it. If you’re roasting the turkey breast, leave the skin on it (if that is an option) through the cooking. We would spread a bit of butter or oil on the breast before putting it in the oven, cook it at a relatively low temperature (325°F), and baste it occasionally.