A Link to the Past (GBA)
The basket can be acquired from Q. Bumpkin, brother of the Twin Lumberjacks. If the save file has data for the companion game, Four Swords, in which Link has collected 10 medals, Bumpkin gives Link the basket. This item is part of the Riddle Quest, and is needed to complete it - Link must use the Bug-Catching Net to pick up items to place in the basket.
The reward for completing this quest is the Hurricane Spin.