Poipu Beach is a public beach and is about 40 minutes from the port in Nawiliwili. We rented a car at the airport (about 5-10 minutes away from the port) and from there it took us about 1/2 hour.
I dont know about the tender dock as I have never cruised into Kauai. The beaches near Puapu are wonderful but can be rough. Check out shipwreck beach. its a surfers beach but beautiful.
Actually, you will dock in Nawiliwili Harbor, not tender. There is a great beach within easy walking distance right in front of the Marriott Hotel. Poipu Beach is on the south side of the island and is blessed with great weather (sunshine) and great beaches. If you rent a car be sure to stop in Koloa Town just outside of Poipu Beach for shopping, etc.