In a medium bowl, combine onion, garlic, barbecue sauce, gin, vermouth, olive brine, and basil.
Sprinkle each steak with salt and pepper, and place in resealable sandwich bags. Divide marinade between each bag, and seal. Ensure that marinade is evenly distributed around steaks. Refrigerate for 2 to 24 hours.
Preheat grill for medium-high heat, and lightly oil grate.
Remove steaks from bags, and discard marinade. Grill steaks for 7 to 8 minutes per side, or to desired doneness.