For the simple syrup:

Make the simple syrup by heating the sugar and water in a saucepan over medium heat. Do not boil. Stir until the sugar dissolves, about 3–5 minutes, then remove pan from stove. Cool and transfer to a clean bottle or jar. Cover and refrigerate. 

For the cocktail:

Make the cocktail by muddling the 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.