Was the narrator a bank robber?
If you are asking about the omniscient narrator of O. Henry's short story, the narrator is an objective narrator who keeps a distant proximity and who does not intrude into the story with subjective (i.e., personal) comment. Consequently, it is not possible to know if the narrator is a bank robber since we can know virtually nothing about the narrator.
A contrast to this objective, distant type of omniscient narrator is the subjective, close type of omniscient narrator who does intrude comment into the story. Two examples of this contrasting type of close, intrusive narrator are the narrators of Jane Austen and Thomas Hardy, who don't hesitate to comment and introduce emotional reflection into the narration, making it, in contrast to O. Henry's narration, subjective, not objective narration.
The consequence of using this type of narration is that the reader knows no facts about the narrator, including whether s/he is a bank robber. Contrast this with the narrator in Fielding's Tom Jones about whom much can be both learned and deduced.
The objective, distant narrator of O. Henry's story reports factually about what characters do, say, think and feel. Even when the narrator reports on thoughts and feelings, it is done without subjective intrusion; it is done with distant objectivity. An example of the factual reports this narrator provides describes a period of time following Jimmy's release:
Two weeks after that, a safe in Logansport was opened. It was a new kind of safe; it had been made, they said, so strong that no one could break it open. But someone did, and took fifteen hundred dollars.
Two examples of how this narrator reports thoughts and feelings with the same distant objectivity occur in an exchange with (1) the "chief prison officer" and with (2) Jimmy's benefactor, Mike:
“Me?” said Jimmy in surprise. “I never broke open a safe in my life.”
“Oh, no,” the chief prison officer laughed. “Never."
Mike enjoyed these words so much that Jimmy had to take a drink with him. Jimmy had some milk.
O. Henry employs an omniscient narrator, about whom no information is given. Perhaps, then, the intent of this question is whether the protagonist is a bank robber?
This response will address the question as pertaining to the protagonist, Jimmy Valentine.
In the course of the narrative of "A Retrieved Reformation," Jimmy is referred to as "a safe-cracker." After his release from prison, Jimmy resumes his illegal occupation. While his exploits are alluded to, they are simply referred to as "safe-burglary" and not specifically as bank burglaries. However, there is one mention of a bank:
Then an old-fashioned bank-safe in Jefferson City became active and threw out of its crater an eruption of bank-notes amounting to five thousand dollars.
After Jimmy Valentine arrives in Elmore, Arkansas, he falls in love and decides to reform his life. With an ironic turn to his life, however, a little girl is locked into the new bank safe where Jimmy's future father-in-law is the president. Jimmy uses his skill in order to break the safe and save this child. So, it can be assumed that his other jobs have probably been ones of "cracking" bank safes.
Since the story does not imply or state who the narrator is, we can assume that the narrator is the author, O. Henry. O. Henry was never a bankrobber, but he once was accused of embezzling funds, which was probably attributed to his carelessness. This is all O. Henry has ever done to break the law.