Heat the sugar and water in a saucepan over medium heat. Do not boil. Stir until sugar dissolves, about 3 to 5 minutes, then remove the pan from the stove. Cool. Transfer the syrup to a clean bottle or jar. Cover and refrigerate.
Make the cocktail:
Muddle basil leaves with lemon juice and simple syrup. Add ice and gin and shake. Double-strain into a chilled cocktail glass with ice. Garnish with a basil sprig.