To select an alternate costume, hold a direction of the control pad when you drag the player onto the ring. Different directions indicate different costumes.

Black Mage's alternate costume - Finish Hard Rainbow Cup (bronze prize)
Daisy's alternative costume - Finish Flower Cup (silver prize)
Flyguy's alternative costume - Finish Hard Star Cup (silver prize)
Ninja's alternate colors - Finish Hard Rainbow Cup (gold prize)
Peach's alternative costume - Finish Mushroom Cup (silver prize)
White Mage's alternative costume - Finish Hard Rainbow Cup (silver prize)
Yoshi's alternative costume - Finish Hard Flower Cup (silver prize)