Beat the game and save it. Then, select the saved game and hold down both the L and R BUTTONS, then enter A. The green sporting super mario bro is unlocked.

Submitted by Toady

I can confirm that you do not need to clear the game to play as Luigi. Just hold L + R and press A on the file you want to play, and you will be Luigi, regardless if you finish the game or not.

Submitted by Henry Hankchill