For the syrup: Boil 1 cup water, sugar and basil over medium heat in a small saucepan. Reduce the heat and simmer, stirring occasionally, until the sugar has dissolved, about 5 minutes. Remove the pan from the heat and allow the mixture to cool, about 15 minutes. Strain before using.

Place the syrup, peaches, rum, liqueur and lime juice in a blender. Blend until smooth. Pour into ice-filled glasses and garnish with lime wedges and basil sprigs.