Put the Mahi Mahi in a large resealable bag. Add the teriyaki sauce and dried basil to the bag and mix it well so the fish is completely covered.
Refrigerate the marinating fish for about 1-2 hours.
After the fish has marinated, remove the fish from the marinade and set aside, reserving the marinade. Pour 1/2 cup of the reserved marinade into a bowl and add 1/2 cup of mayonnaise. Mix very well.
Heat a skillet or a grill to medium high heat. Take each piece of fish, dip in the mayonnaise mixture and put it immediately on the grill or skillet. Cook for 4-5 minutes on each side.
Serve with your favorite vegetable and a side of rice.