What can I do with this basket on the dock?

Solution 1:

When you get access to Tortimer's Island, you're not allowed to bring your items from your town to the Island. The only way you can bring things back from the Island is to make use of Items Drop-Off Box next to the counter.

Anything you put in this box will appear in the basket back on your dock for you to pick up. It holds 40 items, and while it tells you to pick them up "as soon as possible", it's not yet known if there's an actual time limit to how long items will stay in there.

Also, make sure you put anything you want to keep in the box before you leave, as if you don't, anything in your pockets will automatically be sold to Leila, who will only give you 1/20th of the bell worth for your items.

Additionally note that the basket persists with your character; if you leave items in the basket in a friend's world then return to your own world the items will be in the basket in your town waiting for you.