Combine the balsamic vinegar and sugar in a small saucepan over medium heat. Bring to a simmer, stirring occasionally, until the sugar is dissolved. Turn off the heat and set aside to cool.
Starting with the watermelon squares, push the watermelon to the very tip of the skewer. Then skewer a basil leaf, then a tomato half. Continue with another watermelon, basil leaf, and tomato half. Place the skewer on a serving platter so it stands upright, using the lowest watermelon square as a base. Continue with the remaining skewers.
Drizzle the skewers with the reserved balsamic syrup and the olive oil. Sprinkle with coarse salt. Serve.