Hi janfred. I live not far from Newport and, like you, have been there several times. One of our C/NE cruises made a port call there, and I stayed on the ship! But, I just asked my husband where the tender dock was, and he said it was in an area adjacent to the Museum of Newport History. ("Near the yachts," he said, which wasn't much help at the beginning.
As for what to do besides the mansions, there is the Cliff Walk, if you are so inclined, that is along the coast right behind the mansions. Downtown, within a reasonable walking distance of the tender port (assuming my husband was correct) are the Touro Synagogue, a few museums, and a quaint shopping area.
I'm happy to find out more for you.