In a large saucepan, combine sugar, peaches, basil and water; bring to a boil. Reduce heat; simmer, uncovered, 5 minutes. Remove from heat; let stand 30 minutes. Discard basil; stir in lemon juice. Refrigerate until cooled completely.
Place peach mixture in a blender; cover and process until blended. Strain into a pitcher; add additional cold water to reach desired consistency. To serve, fill glasses with ice. Pour peach mixture halfway up glass; top with club soda or, if desired, champagne. Garnish with additional basil.