Add the cherries, lime juice, and simple syrup to a blender. Blend until very smooth.
Set a strainer over a jar, and pour the cherry mixture through. You can use a spoon to stir it around in the strainer, but don’t press the pulp, down. You should about 1 cup of cherry lime juice when you’re done.
Set up 4 rocks glasses. Add a basil leaf to the bottom of each one, then fill with ice. Give the glass a few shakes to bruise the basil, and release its essence.
Pour a ½ cup of seltzer into each glass. Add ¼ cup of cherry juice. Stir gently to mix the soda. Serve immediately.