Melissa, all the beaches in the US Virgin Islands are public. I don't know one that is right near the cruise ship dock. The closest are on the Marriott properties, with a man-made one recently created for the new Marriott timeshare resort.
Lindqvist is on the east end of the island. The property has recently been bought by the USVI gov't and is being managed by the same group that manages Magen's Bay. There are fewer amenities on Lindquist, and also a lot fewer visitors. I'm sure as the improvements are made it will become more popular, but for now it's a beautiful, quiet almost pristine beach.
If you type "Lindquist St. Thomas" in youtube.com you can see a couple of videos taken there.
Another source for information on all the St. Thomas beaches is
Virgin Islands Now Hit the St. Thomas drop down menu and click on "beaches" for some good descriptions of the many differnt beaches on the island.