How would you characterize the cops in "The Cop and the Anthem"?

Several police officers in O’Henry’s story can be characterized as incredulous, careful, and efficient.
When Soapy throws a rock through a shop window, a nearby officer does not arrest him. The officer is incredulous: he does not believe that the culprit would hang around.
When Soapy makes inappropriate comments to a woman on the street, the officer recognizes, as Soapy did not, that the woman is a prostitute. The officer is careful not to intervene. While today’s reader cannot know the exact laws of the time regarding solicitation, as the man had not engaged the prostitute’s services, no crime had been committed. A policeman who regularly patrolled a neighborhood would probably know a woman working that area and might be reluctant to charge her.
Finally, the officer who does arrest Soapy is doing his job efficiently. Soapy is in fact a vagrant, and the officer takes him into custody.

O. Henry uses the police officers in "The Cop and the Anthem" to highlight the difference between laws and their enforcement.
As the hero Soapy commits several valid offenses, a series of "cops" apply their own discretion, authority, and prejudiced perception of the facts. First, the officers fail to acknowledge his crimes, then arrest him for the appearance of a crime when he's legitimately innocent.
Homeless Soapy's goal is obtaining a warm and well fed jail cell for the Winter, so he intentionally begins breaking laws. Each particular offense Soapy commits is actually a crime but for a variety of reasons no charges are brought.
Soapy begins his crime spree by tossing a rock through a plate glass shop window. Because he stood by until the law arrived, and admitted the crime, the officer didn't believe him, arresting instead an innocent man who just happened to be suspiciously running further down the block.
His next crime, eating in a restaurant without the money to pay, illustrates how the cops can be left out of the picture entirely. The restaurant workers apply their own vigilante justice and serve as judge, jury and executioner. They physically tossed Soapy out on the sidewalk and considered justice well served.
Soapy's attempt to insult a lady enough to get arrested backfires when she turns out to be a prostitute and willing to accept his advances. Similarly his theft of a man's umbrella goes unreported as the fellow had stolen the umbrella himself.
Some crimes are intentionally ignored. Soapy's attempt at mock "drunk and disorderly" behavior in front of an officer is dismissed as boyish exuberance and not worth prosecuting, much the same way that marijuana laws and sodomy laws go generally unenforced today.
Once Soapy wanders into a church and resolves to change his ways, vowing to seek employment and raise his status, he arouses the false suspicions of the story's final cop.
Because Soapy looked so out of place, the officer believed he had to be up to no good and arrested him. African-Americans today face similar false accusations for nothing more illegal than visiting a typically white establishment.

Soapy is keen to get arrested by any policeman. He commits several of his misdemeanors in plain sight of a uniformed cop but fails to have his wish fulfilled. Even as the story begins, Soapy is committing one or two misdemeanors by sleeping on a park bench. He could be arrested every night for vagrancy. The cops who patrol Madison Square must see him there but leave him alone as long as he is gone by morning. When he hurls a cobblestone through a plate-glass store window he tells the cop he is the guilty party, but the cop does not believe him and does not want to have a lot of trouble hauling a prisoner to the station only to have him deny everything. He molests a woman in view of a cop who is standing nearby watching him. He acts drunk and disorderly in front of another cop who thinks, or perhaps pretends to think, he is a college boy celebrating a football victory. He steals an umbrella with a cop standing a short distance away but the owner is afraid to press charges because he stole the umbrella himself.
The plain fact is that cops past and present do not like to make arrests. Arresting people takes up a lot of their time. They may have to write up reports in legalese, and in many cases they may have to appear as witnesses in court on their own time, with or without compensation. Judges do not want the police to bury them under heavy case loads. The public does not want to pay higher taxes to create more judges and more court houses. No doubt the police could arrest ten times as many people on one charge or another, but the courts cannot handle the cases and the jails cannot accommodate all the prisoners. Only a small percentage of crimes actually get punished. Nearly everyone commits crimes or misdemeanors at one time or another. As Hamlet asks: "Use every man after his desert And who shall 'scape whipping?" Or as Jesus says to the mob who want to stone a woman to death for committing adultery: "He that is without sin among you, let him first cast a stone at her" (John 8.7). Which of us could cast that first stone? Another of the truths, or themes, of "The Cop and the Anthem" is that cops serve mainly as symbols of the law, and the law itself is largely symbolic because if all the laws were strictly enforced half the population would be in jail and the other half would be guarding them.
An article in Wikipedia quotes the following statistics:
According to the US Bureau of Justice Statistics (BJS), 2,266,800 adults were incarcerated in US federal and state prisons, and county jails at year-end 2011—about 0.94% of adults in the US resident population. Additionally, 4,814,200 adults at year-end 2011 were on probation or on parole. In total, 6,977,700 adults were under correctional supervision (probation, parole, jail, or prison) in 2011—about 2.9% of adults in the US resident population.