Combine 2 1/2 cups cherries, sugar, water, and 3 large basil sprigs in a medium saucepan. Bring to a boil over medium-high, stirring until sugar is dissolved. Remove from heat, and cool completely (about 1 hour).
Pour syrup through a fine wire-mesh strainer into a pitcher, discarding solids. (Yield should equal about 1 cup.) Cover and chill until very cold, about 2 hours.
For each serving, pour 2 Tbsp. of the syrup over ice in a 12-oz. glass, and stir in 3 Tbsp. vodka. Top each with about 1 cup chilled kombucha. Garnish, if desired.