In a small saucepan, bring sugar and water to a boil. Cook and stir until sugar is dissolved. Place half the basil in a small bowl. With a pestle or wooden spoon, crush basil until its aroma is released. Stir into sugar mixture. Remove from heat; cool completely. Strain; refrigerate until cold.
Place 2 cups crushed ice and remaining basil in a 4-qt. pitcher. Using a muddler or a wooden spoon, press basil leaves against ice until their aroma is released. Stir in basil syrup and soda. Serve over crushed ice in tall glasses; squeeze lemon wedges into drink.