We don’t know much about Asenath, except that she was Egyptian and the daughter of daughter of Potipherah priest of On. On was a city in Lower Egypt, bordering land of Goshen. It was the centre of sun-worship.
Asenath was given to Joseph in marriage by Pharaoh himself. The name Asenath means, “belonging to the goddess Neith.” Neith was an ancient goddess of war and weaving.
According to apocryphal narrative Joseph and Aseneth, she is a virgin who rejects several worthy suitors in favor of Joseph, but Joseph will not have a pagan for a wife. She locks herself in a tower and rejects her idolatry in favor of Joseph’s God, Yahweh, and receives a visit from an angel who accepts her conversion.
A ritual involving a honeycomb follows. Bees cover her and sting her lips to remove the false prayers to the pagan gods of her past. Joseph now consents to marry her. She bears him two sons, Manasseh which means, “For God has made me forget all my toil and all my father’s house.” and Ephraim which means, “For God has caused me to be fruitful in the land of my affliction.”
Rabbinical literature, Asenath is believed to be the daughter of Jacob’s daughter Dinah and Shechem, the man who raped her. The story says that she was taken by angels to Egypt, where she was adopted. In some versions she is identified by a special plate which Jacob placed around her neck, bearing the name of God and/or the story of her conception.
It is clear from the true account of the story of Joseph and Asenath found in the written Word of God, that God didn’t have a problem with Joseph marrying her. It is very likely that she embraced his faith or at least, unlike Solomon’s exotic wives, she didn’t lead Joseph into idolatry. I doubt that she would have succeeded even if she had tried.
Sources: Wikipedia; Blue Letter Bible