Prepare the simple syrup: In a saucepan, heat the sugar and 1 ¼ cups of the water over medium heat, stirring frequently until the sugar has dissolved. Cool to room temperature.
In a pitcher, combine the simple syrup, lemon juice, beet juice, and remaining ½ cup of water. Stir to combine. Taste, and add more water if desired. Serve over ice and garnish with basil leaves.