Problem 1: By its definition libertarianism always favors smaller government.
It is popular to denounce "big government" in politics and perhaps government is too big. But libertarianism always answers the question of how big government should be with, "smaller." Indeed I believe government is too big in today's society, however was it too big in Dickensian London? According to a libertarian the answer is "yes." Yet the truth is for all their denounciations of big government most people on the center-right still favor bigger government than in the year 1900, but by its definition libertarians want to make government even smaller yet.
Problem 2: Libertarianism is relativistic, yet all law reflects morality.
Many libertarians (and some liberals) say you "can't legislate morality" yet by its definition all law is a form of legislating morality. In the end somebody's morality will be legislated, the question is not if, but the question is whose morality. Obviously we can avoid sectarian morality, for instance, preventing right-wing religious fanatics from legislating their agenda, but sooner or later moral questions will always work their way into the law.
Problem 3: Libertarianism cannot answer the problem of children without violating its own code.
One problem with libertarianism is the state of minors. Indeed libertarianism often talks of "consenting adults" and also "non-aggression axioms." Yet who could argue that children are able to enter into contracts or hold adult responsibilities? Few would, yet they are still human and with the non-aggression axiom you are violating their rights by restricting their freedom for their own good and allowing parents to do so, the paradox being that other adults could exploit children even more if they did not. For instance we "initiate force" against children by preventing child labor, yet it would be easy for children to be exploited by child labor and work in coal mines.
In the end we solve this by creating an age of majority, or more realistically, phasing in of various ages of responsibility. There can be legitimate debate about where the correct age of majority is, how old people should be before they can work, vote, sign contracts etc. In the end I think it would probably be somewhere between the late teens and early twenties which is where society sets it. Below or above this too much would defy logic, but like or not it is going to have to be arbitrary at some point. Why are 18 year olds adults? Some could say 17 year olds should be considered adults too, others would say 18 is too irresponsible and maybe it should be 19, but like it or not sooner or later it is arbitrary within a certain framework.
Problem 4: Most "libertarian" politicians are not libertarians. The reason being is most people are unwilling to accept full bore libertarianism.
This applies to so-called "libertarian" Republicans. Even so, these "libertarian" Republicans often support immigration restrictions and some elements of social conservatism, often reverting to a "states' rights" stance on social issues. For instance you don't see these so-called libertarians unapologetically arguing for the legalization of heroin and prostitution and for open borders. Yet in the end this is libertarianism. Short of this and you restrict liberty. Still they are somehow afraid to admit what it really is. You even have so-called "libertarians" arguing for piecemeal reforms like "personal accounts" in Social Security. Again, not libertarians.
Case in point you can build a coalition around legalizing marijuana, yet by definition a libertarian, being always for greater liberty, cannot escape the logical conclusion of favoring legalization of even harder drugs yet. Still how can anybody sit and deny the reality of supply and demand? You create a cost to produce these drugs by waging war on them, not having those risks would make them cheaper and by definition cheap things usually become plentiful. The argument around marijuana is not for me an argument about whether or not the drug war is "working" because the truth is most casual users get a slap on the wrist. In the end the issue is not whether marijuana legalization will make its use more plentiful (it certainly will) but whether or not the problems of marijuana are so trivial that they are no cocern of the government (a legitimate point). But how can anybody look at hard drugs and call them trivial?
Problem 5: Taken to its logical conclusion libertarianism leads to anarcho-capitalism and anarcho-capitalism is unworkable.
This is a point anarcho-capitalists are often proud to point out and in my opinion minarchists are merely doing mental gymnastics to justify otherwise. If you can't choose your police, your military and your government on a free market how is that anything less than a violation of the non-aggression principle? Indeed logically a true libertarian has to lead to "anarcho"-capitalism. The problem is anarcho-capitalism differs from historic anarchism in a few ways and is unworkable.
Anarchism historically did not ever propose multiple laws over one territory. The idea of anarchism is simply the abolition of hierarchy, not law itself. Thus the people rule as directly as possible. Anarchist theory has held varying degrees of thought such as radical direct democracy and syndicalist economics. However anarcho-capitalism differs in that it isn't the people directly ruling in a directly democratic commune, but various agencies resolving disputes over one territory. These dispute resolutions are often proposed to be peaceful, but in the end there will always be disagreements. I don't see how it would not degenerate over time into something resembling gang warfare. Anarcho-capitalism doesn't work because it tries to square the circle by having a society of law in which there are multiple private laws followed in one single area.
Problem 6: Libertarians are often right, but then go too far.
Case in point zoning laws. Perhaps it is a good thing to let development move organically and not try to restrict everything into low and high density, commercial and residential, promoting government enforced suburbanization. However libertarians would move beyond this and do away with building codes altogether. Yet we see how societies without building codes are prone to disaster (Great Chicago Fire of 1871). Yet for the libertarian stacking up wooden shanties with faulty electrical wiring is "property rights" even if it in endangers the property of others.
Problem 7: Property is a creature of society. There are no "natural rights" to property.
Indeed moving onto property rights, property rights in and of themselves exist within a legal framework. Someone has to define whose property is whose and how it works out. I am not saying abolish private property, absolutely not, but as a creature of society there cannot be a natural and philosophical right to invioable private property rights that exists in the same category as, for instance the right to life. This doesn't mean property rights should not be protected, merely that we have to be realistic about the origin of property rights.
Problem 8: The welfare state and social cohesion.
Perhaps our social safety net does too little to encourage work and too much to encourage sloth. Again one can apply common sense principles without going no holds barred. Yet there are indeed slothful people in our society, slothful and entitled people. Libertarians should by definition favor immediate abolition of the welfare state. Yet cut off from succor the natural human evolutionary drive kicks in. Indeed the welfare state has allowed many people to be fat and happy instead of being in survival mode, yet it takes weaning off over time. Indeed the natural reaction to crisis is survival by any means neccessary. It makes evolutionary sense that if a welfare state is available some people will naturally mooch off of this if indeed they desire little more than this, whereas others are incentivized to work because they seek to enjoy a standard of living beyond mere survival. We all have what we value. Yet most humans value survival and if survival is up in the air there no doubt you would see more crimes created in the name of survival.
Perhaps it violates property rights to tax for welfare, yet if people in desperate survival mode go out and begin to commit various crimes for survival it might violate property rights more significantly. Indeed a burned down business is much worse than filling out a form to pay taxes.
Problem 9: The gold standard and monetary policy.
This is one I just don't get. At the end of the day they can talk about how fiat money has no "real value" beyond what humans give it but gold is not much better. Yes gold has some electrical properties and malleability which make it a good industrial metal. Yet most of the value of gold is not from its industrial use, but merely from what culture has ascribed to it. Gold would be valuable without the cultural attachment to it as an object of beauty (imagine we lived in a world of purely utilitarian concerns) but it would still be significantly less valuable. Indeed golds value comes from scarcity of supply and demand, fiat money gets its value the same. In the end it seems the only real gripe about fiat money they have is it comes from (gasp!) government.
Problem 10: The seamless garment of liberty.
I am no neo-con and do not believe that the west can simply will democracy and freedom on the world, yet there is no doubt that various alliances of democratic nations such as NATO and the United Nations have at times intervened when times are dire. Indeed there is no doubt that the interventions in Bosnia and Kosovo stopped genocide short before it could grow even worse. This is because society has grown sensitive to genocide after matters such as the Holocaust, the Armenian genocide and the Japanese occupations of other Asian nations during WWII. Rightly we said "never again." We did fail at times (Cambodia, the Second Congo War, Rwanda etc.) to step in but in many others it worked.
Indeed even if one accepts minarchist libertarianism the question is why should we only worry about the liberty of our fellow countrymen? Is not liberty a seamless garment that should logically apply as a worldwide thing. Is it not desirable where it is feasible for NATO and others to intervene in times of great humanitarian crisis? Indeed borders are merely historical matters. Nobody has explained yet why an invisible line should mean that I worry about the liberty of those within it more than those outside of it.
Please note the same logic also applies to the interest in neo-Confederate ideas found in some sectors of libertarianism. How was it a greater violation of civil liberties for the North to intervene than chattel slavery was?