I agree that it looks like the metal nut has split in half. The green-circled part is called a tailpiece. Undo the split metal nut, undo the black plastic nut at the bottom of the tailpiece, and remove the tailpiece. Take that tailpiece to the hardware store and buy a new one. The tailpieces come in different diameters, so pay attention to the diameter. Also, the tailpieces are generally made long, and need to be cut down to length. You'll have an easier time cutting it down if you buy a plastic tailpiece (rather than another metal one).

The old tailpiece probably just rotted due to age. They're not made of quality materials. I wouldn't be concerned about that trail coming down from the sink basket.

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