Create foil bowls by curling up the edges of aluminum foil pieces to hold liquid. Place salmon fillets in the foil bowls.
Position the foil packets on a flat baking sheet or cutting board.
In a small bowl, combine vegetable oil, soy sauce, Worcestershire sauce, lemon juice, ginger, honey, and dried basil. Whisk until well combined.
Pour the marinade over the salmon fillets, making certain each piece is well coated.
Let the salmon marinate for 15 minutes while preheating the grill.
Preheat grill to medium-low heat.
Carefully slide just the foil with salmon onto the grill grates.
Cover with the grill lid and cook for 10 minutes per inch of thickness, approximately 20 minutes total.
Test for doneness by checking if the salmon flakes easily with a fork but isn't dry.
Remove from grill and serve immediately.