In a medium bowl, toss together mango, cucumber, basil, lime juice, and olive oil. Season relish with salt and pepper.
Heat a cast-iron skillet or grill pan over medium-high. Using a paper towel, blot fish dry. Season both sides with salt and pepper. When pan is hot, add vegetable oil, then gently add fish, working in batches if necessary (do not overcrowd pan). Cook fish, undisturbed, until fillets release easily from pan, about 6 minutes. Flip and cook until opaque throughout, 4 to 6 minutes. Serve fish with relish.