Heat the sugar with 1 cup of water in a medium saucepan over medium heat, stirring occasionally, until the sugar dissolves. Remove from the heat, add the basil, and let steep until the syrup cools to room temperature. Discard the basil.
Puree the strawberries in a food processor. Add the cooled basil syrup, pulse until combined, then pour the mixture into a large baking dish.
Freeze until the mixture begins to harden around the edges, about 30 minutes. Stir with a fork, breaking up some of the pieces, then return dish to the freezer for 2 hours, scraping the surface with a fork every 20 minutes to create a flaky texture. Transfer the granita to a covered container and freeze for 20 minutes more, or until ready to serve. To serve, divide among 6 serving cups and garnish each with a sprig of basil.