Combine 2 cups corn kernels, eggs and sour cream in a blender; season with salt and pepper and pulse to blend until smooth. Transfer to a bowl with remaining corn kernels, scallion whites, cornmeal and flour, and stir just until combined.
Heat 1/4 cup olive oil in a large skillet over medium-high heat. Working in two batches, spoon the corn batter into the skillet and pat to flatten into small cakes (about 1/4 cup per cake). Cook, flipping once, until golden on both sides, about 6 minutes per batch.
While cakes cook, stir together remaining 1 tablespoon olive oil and vinegar; add crab, tomatoes and 2 tablespoons scallion greens to bowl and season with salt and pepper. When all the cakes are cooked, toss crab and tomatoes with basil and vinaigrette. Serve cakes topped with crab salad.