Prepare the grill (medium-high heat). Whisk the oil, mint, lemon juice, basil, and garlic in a medium bowl to blend. Season the lemon and olive oil mixture with salt and pepper, to taste.
Brush the swordfish steaks with 2 tablespoons of the lemon and olive oil mixture. Grill the steaks until just cooked through, about 3 minutes per side (depending on thickness of steaks). Transfer the steaks to plates. Spoon the remaining sauce over and serve.