As a ship sails a steady course the North Star sinks in the sky In what direction must the ship be sailing?

It is moving South. At sea you could measure the angle between the horizon and Polaris with a sextant. This angle is approximately equal to your Northern latitude. There are some mathematical corrections one can perform to get a more precise answer, but the further south you sail, the lower Polaris will become. Eventually, you will not be able to see it anymore. Polaris is not visible from the Southern Hemisphere and there is no South Star. The ship is in the Northern Hemisphere, moving South.