In a small bowl mix oil OLD BAY lime juice honey and basil. Place tuna steaks in large resealable plastic bag and add marinade. Turn to coat well.
Marinate in fridge 20 minutes (longer for extra flavor).
Transfer fish to grill and discard any leftover marinade. Broil or grill fish over medium heat 5 to 6 minutes per side or until desired doneness. (Tuna is more moist when its center is still rosy-pink after cooking.)