What does Prospero do to bring the mariners to the island in The Tempest?
Prospero used magic to bring the mariners to the island.
To explain that in a bit more detail, Shakespeare's play begins with a ship facing a storm, the "tempest" that gives the play its title. The storm is so intense that the sailors are afraid it will sink or destroy the ship. In fact, there's a tremendous sound in Act I, Scene 1, which makes the sailors think they really are about to die.
This was all orchestrated by Prospero, who sent Ariel, a spirit in his service, to shape the storm. Ariel made the wind, lightning, and thunder. Ariel also made sure everyone made it to the island safely, so you could say the combination of Prospero's magic, his desire for revenge, and his sense of responsibility brought the sailors to the island.