1. Combine first 3 ingredients in a small saucepan; bring to a boil, stirring until sugar dissolves. Stir in basil; remove from heat. Cover and let stand 30 minutes.
2. Place cherries in a food processor; process until smooth. Add sugar mixture, grapefruit juice, and lime juice; process until well blended. Strain cherry mixture through a fine sieve over a bowl; discard solids. Pour cherry mixture into the freezer can of an ice-cream freezer; freeze according to manufacturer's instructions. Spoon sorbet into a freezer-safe container; cover and freeze 1 hour or until firm.