This story involves the kind of situation that must have occurred thousands of times over the years all over the world. A cop runs into an old friend who is committing a crime or is wanted for a crime, and he has to decide whether to arrest him or let him go. We can understand how Jimmy might feel, but he has a duty to uphold the law, and the law makes no exceptions for friendships. There should be no question about what Jimmy should do, and no hesitation.
O. Henry has softened Jimmy's problem in order to keep the reader from blaming him too severely for betraying an old friend. For one thing, Jimmy does not make the actual arrest. That would be a painful scene. Instead Jimmy turns the "collar," the arrest, over to another officer--so Jimmy does not really make the arrest. Furthermore, the plain clothes man tells "Silky: Bob:
"Chicago thinks you may have dropped over our way and wires us she wants to have a chat with you."
This sentence suggests several things. Bob is not wanted for trial or for prison-breaking; he is apparently only wanted for questioning. It may be that he is not even guilty of whatever offense it is that they want him for questioning about. Since Bob is being arrested in New York, he cannot be summarily shipped off to Illinois but would have to be extradited. Otherwise the Chicago authorities would have to send someone to New York to question him. He seems to have plenty of money, so he could fight extradition and make the process so difficult that Chicago might just give up on him. The words "a little chat" might be euphemistic, but they suggest that there would be a lot of grilling which did not result in an indictment. After all, they don't call him "Silky" Bob for nothing. He may be able to talk his way out of whatever jam he is in. Even the fact that O. Henry has the arresting officer refer to Chicago as "she" has a softening touch to it. So everything that makes Bob's predicament seem less drastic also makes Jimmy's betrayal seem less reprehensible.
Would "Silky" Bob have traveled all the way from Chicago to New York City if he had known that his friend Jimmy Wells had become a cop?
The short story 'After Twenty Years' by O. Henry shows us very troubling themes and proves that the author is a classic writer because those troubling themes still disturb us today - they are thought-provoking and make us stop to consider the ways in which we live our lives. The themes include morality or 'religion' or belief as these themes affect believers in the way they make decisions about the way forward in any given circumstance, because such citizens are people of faith. Other themes are guilt, innocence, time, duty, conscience and friendship.
Jimmy has to consider what is right, but you could try considering how he might do the right thing and help his friend at the same time - it's not an impossible task. For example, if he loved his friend, he might want to help him lead the most perfect and happy life possible. That might entail apologising for unfair things he did that hurt people, or impoverished them. He could help his friend to 'fess up' and make recompense by having a long heart to heart with him, getting him to see the error of his ways. In short, I am suggesting that he tries to convince his friend to give himself up.