When Jason crosses the river with Hera on his back, it is a rite of passage. The young man is not only crossing a body of water, he's also crossing into another stage of his life. Jason's life with his mentor, Chiron, is over, and now he has to apply everything he's learned out there in the real world. The torrential river is the first challenge of the many that Jason will have to face, and the fact that he passes the test shows that he just might have what it takes to be a great hero. He starts into the river a boy; when he emerges on the other side, he's a man.
The crossing of a river is often used to symbolize passing from won state to another. Another example from Greek mythology is the crossing of the River Styx, which divides the land of the living from the world of the dead. In order to cross this stream, souls must pay Charon, the ferryman, to take them across in his boat. (You have to pay for that?) So, the crossing of the River Styx represents passing from a state of living to a state of death.
Another famous river crossing from world mythology is the crossing of the River Jordan, which took the Hebrews into the Promised Land. In the Judeo-Christian belief system, this crossing represented a promise made by God that the Jewish people would one day find a homeland. To the Hebrews, crossing the river transformed them from a wandering people to a people with a home. It also symbolized that they were sacred to God.