Combine flour, basil, salt and pepper in large resealable plastic bag; shake to mix well. Add chicken. Seal and shake to coat; set aside.
Melt 2 Tablespoon of the butter in a large nonstick skillet over medium-high heat. Add chicken breasts. Place onion around the edges of chicken. Cook chicken for 4 minutes per side to lightly brown; stir onion occasionally.
Pour broth and vinegar around chicken. Add cherries to pan. Cover; reduce heat and simmer 4 minutes until no longer pink. Remove chicken to platter; cover to keep warm.
Increase heat until mixture bubbles; cook 2 to 4 minutes to reduce liquid, stirring occasionally. Reduce heat to low; stir in remaining 1 Tablespoon butter until blended. Serve chicken with sauce.