Whisk together the sake, miso paste, mirin, soy sauce, and brown sugar in a baking dish. Add the salmon, and allow to rest for 5 minutes before flipping the fillets over in the marinade to coat. Cover and refrigerate for 2 hours.
Bring salmon to room temperature, about 15 minutes, then remove salmon from the marinade, and shake off excess. Discard the remaining marinade.
Heat a large skillet over medium-high heat. Arrange salmon in the skillet, and cook until the fish flakes easily with a fork, about 4 minutes on each side. Transfer the fish to a plate and garnish with basil.