Directory:
Great Cruz Bay
There is a large but crowded mooring field and most of the balls are private.
There’s nothing a mooring owner can do about your anchoring on top of a mooring as the mooring owner only has title to the ball and ground tackle, not the space on the water.

Some moorings are owned by landlords on the shore and even if they are not on the island, some have caretakers who will dinghy out to ask you not to use the mooring.
The beach left of the main buildings of the Westin Hotel is good for dinghy landing. You can take a taxi from the hotel for the short ride into Cruz Bay.