In a large bowl, combine the basil, lime juice, olive oil, onion, jalapeño and garlic. With a rubber spatula, gently fold in the avocados and season with salt and pepper.
In a large skillet, heat the vegetable oil until shimmering. Season the scallops with salt and pepper and cook over high heat until browned outside but still rare in the center, about 2 minutes per side. Transfer the scallops to a warmed platter.
Add the water and tomatoes to the skillet and cook over moderately high heat, scraping up any browned bits from the bottom, until the water has reduced to 2 tablespoons, about 2 minutes. Remove from the heat. Stir in the balsamic vinegar and season with salt and pepper. Pour the pan sauce over the scallops and serve the avocado salsa on the side.
Notes: Variations: Use the avocado salsa on steak, with boiled shrimp, in a chicken sandwich or with tortilla chips.