Ben Price thinks that Jimmy is still a criminal and wants to arrest him until he saves the little girl, and Ben does not tell anyone who he is and lets him go.

Jimmy does plan to rob the bank, until he meets the love of his life.  Upon seeing her, it is love at first sight for him and she seems to reciprocate because he is so handsome.

A young lady crossed the street, passed him at the corner and entered a door over which was the sign, "The Elmore Bank." Jimmy Valentine looked into her eyes, forgot what he was, and became another man. 

Jimmy changes his name, becomes respectable, and begins to court the lady.  She just happens to be the daughter of the banker, so it makes sense to suspect him of being up to no good.  Alas, this is not the case.  Jimmy risks his new life and his freedom to use his safecracking skills to rescue his only love’s little sister when she accidentally locks herself in the safe.

If Jimmy had not acted, the girl would have died.  Jimmy was sure that acting meant getting outed and arrested.  He would lose his love, and be locked up.  Yet Jimmy’s transformation was genuine, and he would rather risk anything to spare his love pain.

Ben Price realized what Jimmy had done, and the risks Jimmy took.  He also knew why he took them.  He decided to pretend he did not know Jimmy and go on his way, leaving him to a new life as a reformed safecracker.