What is the Objectivists position on punishment for criminals?

The libertarian paper cited by the questioner is in a sense correct -- the Objectivist position and the libertarian one are incompatible. Objectivism would deny that criminals have the right to buy off their victims. Leaving aside the potential for abuse -- organized crime is very good at intimidating its victims into refusing to press charges -- the libertarian position reflects a fundamental misunderstanding of the nature of crime and punishment.

The purpose of the justice system in a free society is to identify instances in which the initiation of physical force has taken place and to punish the initiators. Whether such initiation has taken place is a question which must be answered by an objective process of investigation. This is the purpose and justification of government, according to Objectivism: to place the use of retaliatory force under objective control. Allowing the alleged victim to short-circuit this process of investigation undercuts the objective controls placed on retaliatory force -- not by allowing retaliation where none is warranted, but by preventing retaliation where it is warranted.

Why would this matter?

What is the purpose of retaliatory force? It is not, contrary to the libertarian position, to compensate or 'make whole' the victim. A justice system which only required the criminal to make his victims whole would turn crime into a win-or-break-even gamble. If you steal a car and get away with it, you have the car. If you get caught, you make the victim whole by giving the car back and you're no worse off than when you started. (The notion of making the victim whole only has even superficial plausibility when dealing with crimes against property -- what would it mean to make a rape victim 'whole'? A kidnap victim? A murder victim?)

If the purpose of retaliatory force is not to compensate the victim, what is it for? It is to close the causal circle on the initiator of force. His willingness to resort to force exposes others to destructive consequences -- not only his actual victims but his potential victims. Retaliatory force visits those destructive consequences on the initiator, which is what makes retaliation just. The criminal, by choosing to use force, has placed himself outside the rules which allow society to function as a life-enhancing arrangement for its members. He cannot buy his way back in by paying off the immediate victim because his choice of force as a means of dealing with others poses an objective threat to all.

The libertarian position is a reflection of the subjectivism that infests their entire movement. From the subjectivist standpoint, the only person capable of deciding whether force has been initiated is the subject himself -- nobody else can say he's objectively wrong. The anarchist view that a government, with its monopoly on retaliatory force, is initiating force against anyone who wants a different enforcer is one expression of this. The idea that if the victim says no force was initiated then no force was initiated is another.

(I wanted to add a few words on why the libertarian position seems plausible. Individual consent does play a role in whether force has been initiated. If someone punches me in the face without my consent we have a criminal assault; if he punches me in the face with my consent we have a boxing match. The libertarian then asks, in effect, why it matters whether my consent is obtained before or after the punch is thrown? The answer: in the latter case the puncher is demonstrating that my consent is irrelevant to whether he will hit me or not. We know he is willing to hit me without my consent because he does so without knowing whether I will consent or not, and it is that willingness to use force absent my consent that makes him an objective threat.)

Speaking of using retaliatory force, John Galt said "I use it only to destroy destruction." This is why we punish criminals--to destroy evil in order to protect individuals from further rights violations. "Justice is the recognition of the fact . . . that every man must be judged for what he is and treated accordingly." Galt’s Speech, For the New Intellectual p. 129. If a man is willing to violate the rights of others, he is a threat to all rational men. The intentional rights violator thereby shows that he does not respect the principle of rights and is going to live on the level of an animal rather than as a human; we therefor treat him as one would a dangerous animal.

In other words, the punishment is not designed to somehow "fix" the past crime that occurred, but to protect against future rights violations. Compensation is an important part of justice and should be required by the law, but it is not the same thing as punishment which is in addition to any compensation. Thus, compensating the victim does not remove the need to punish--the criminal who willfully shot another person has not changed simply because the victim now has some money. He is still the same dangerous person who has demonstrated that he disregards the rights of others (i.e., he is, literally, a loose cannon).

Punishment "destroy[s] destruction" in various ways. In severe cases, the criminal is removed from society completely (e.g., death or life in prison). In less severe cases, the punishment may be simply designed to hurt the criminal in some way, making it more difficult for evil to thrive.

What does the author of the article you are reading put forward as the justification for punishment of criminals in general? If his only justification for punishment is the compensation of the victim, then how would punishment accomplish this in the vast majority of cases? If the criminal has nothing with which to compensate the victim, how does throwing him in jail or inflicting some other punishment (e.g., physical pain) compensate the victim? If you take compensation as the justification for punishment, a lot of crimes will have to go unpunished because they are simply not capable of compensation.

Now, if we are not talking about compensating the victim, but rather of using a monetary fine as one means of punishment, then that is a different story. What specific means one uses to punish is open for debate, and I see no reason why a monetary fine might not be preferable to a jail term or something like that in certain cases. The punishment should fit the crime, however (see Greg's article on that).