Who ferries souls across the river Styx to the underworld? Charon does. In Greek mythology, Charon is the boatman who transports the souls of the dead into Hades, the realm of the deceased. He generally carries only those who have received proper funeral rites and can pay his fare.
That payment is remembered in the tradition of Charon's obol: a small coin placed with the dead, often near the mouth or in the cremation urn. Without burial rites or payment, the dead were sometimes said to remain stranded on the riverbank. Virgil's Aeneid gives such souls a long period of wandering before they may cross.
The river is not consistently named across ancient sources. Greek writers commonly associate Charon with the Acheron, while later and Roman traditions often connect him with the Styx. That variation does not change the answer to this question: Charon is the ferryman, not Hades, the ruler of the underworld; Thanatos, the personification of death; or Hermes, who may guide souls toward the underworld.
Charon also appears in stories about living heroes. Heracles and Aeneas are both said to have crossed into the underworld and returned, requiring special mythic permission rather than an ordinary journey of the dead.