In a small saucepan, combine sugar, basil, and ¼ cup water; bring to a boil over medium-high heat. Cook, stirring occasionally, until sugar is dissolved, about 1 minute. Remove from heat; let steep for 30 minutes. Strain basil syrup through a fine-mesh sieve placed over a small bowl; discard solids.

In the container of a blender, combine nectarines and 2 tablespoons basil syrup; blend until smooth. Strain nectarine mixture through a fine-mesh sieve placed over a medium bowl; discard solids.

Divide grenadine among 8 Champagne flutes; divide strained nectarine mixture among glasses. Top with prosecco; stir until well combined. Garnish with nectarines, if desired. Serve immediately.