Ben Price is described as a big man who is strong and tough. This is noted at the end of the story to show that Jimmy Valentine would have no way of escaping from Ben if he were inclined to try to do so.
At the door a big man stood somewhat in his way.
Jimmy already had one tussle with the detective, which he lost. This is shown towards the beginning of the story, when Jimmy returns to his rented room after being released from prison.
There on the floor was still Ben Price's collar-button that had been torn from that eminent detective's shirt-band when they had overpowered Jimmy to arrest him.
Ben Price exists as a character in the story in order to represent the law in general. He is Jimmy's nemesis. Just as Jimmy is the best safecracker, so Ben Price is the best bank detective. Both are experts at what they do. Ben can tell from examining the looted safes that they are the work of Jimmy Valentine. There is no doubt that he could succeed in having Jimmy convicted for all three bank jobs by testifying as an expert witness that they had to be the work of the notorious Jimmy Valentine.
Ben Price investigated the scenes of the robberies, and was heard to remark:
“That's Dandy Jim Valentine's autograph. He's resumed business. Look at that combination knob—jerked out as easy as pulling up a radish in wet weather. He's got the only clamps that can do it. And look how clean those tumblers were punched out! Jimmy never has to drill but one hole. Yes, I guess I want Mr. Valentine. He'll do his bit next time without any short-time or clemency foolishness.”
Ben is a formidable opponent because he is not only strong, intelligent, and experienced, but he is also tenacious. He traces Jimmy all the way to Elmore, Arkansas, which is where Jimmy is living under an assumed name precisely because he wants to get away from Ben Price. So the bank detective has resources besides his strength, intelligence, experience, and tenacity. He must have contacts all over the country. He can put out the word that he is looking for a certain individual and will soon be told where to find him.
Jimmy is the protagonist in the story. Ben Price is the antagonist. It is necessary to have a protagonist and an antagonist who are equally matched in order to have a interesting and sustained dramatic conflict. Jimmy is the best in his profession, so O. Henry created an antagonist who would be worthy of him. Just as Jimmy is determined to reform and stay out of prison, so Ben is determined to catch up with him and put him back into prison. The big surprise comes when Ben has a change of heart when he observes the heroic sacrifice Jimmy makes by exposing himself to his fiancee and her whole family as a criminal when he uses his expertise to release the little girl who is trapped inside the bank vault.