In a large pitcher, combine basil leaves, lime juice, and granulated sugar. Using a muddler or the back of a wooden spoon, mash ingredients together well, making sure most of the sugar is dissolved. Set aside.
Place about one-fourth of the melon cubes in a food processor or blender. Cover and process or blend until smooth. Press puree through a fine-mesh sieve; discard solids. Repeat with the remaining melon cubes, one-fourth at a time (you should have 4 cups total juice). Add melon juice to the pitcher.
If desired, rub a lime wedge around rims of eight glasses and dip rims in superfine sugar. Fill glasses with ice. Pour juice mixture into ice-filled glasses. If desired, add a splash of carbonated water to each glass. If desired, add melon spears or balls to each glass. Serve immediately.