In a bowl, stir together the mangoes, red onion, lemon juice, basil and capers.
Preheat the broiler. Brush olive oil on both sides of each swordfish steak; season with salt and pepper. Place on a foil-lined baking sheet and broil, turning once, until just firm to the touch, 10 to 13 minutes for medium. Serve with the mango-caper relish.