The great thing about being a libertarian is that there is seemingly little conflict between morality and utilitarianism. Liberty is the most moral system, and it also produces the greatest prosperity on net. Of course, non-libertarians would beg to differ.
The moral case for libertarianism is rather easy to make. This isn’t to say that it is easy to convince others.
If you have a sick neighbor that needs money for medical care, would you personally go to another neighbor’s house and demand money to help the sick neighbor? If your neighbor refused to give you any money (even if he is wealthy), would you demand the money at the point of a gun?
Most people, fortunately, would answer “no”. They would not point a gun at anybody unless it was in self-defense. Even if it was for a seemingly just cause, and even if they thought they could get away with it, most people would not point a gun at another person.
However, most people (i.e. non-libertarians) have no problem delegating this same task to the state. While they won’t personally point a gun at their neighbor to help someone else, they will allow the state to do it for them. Sure, the gun usually doesn’t actually come out, but your neighbor knows he has to pay his taxes and abide by the laws or else the guns will eventually come out to drag him to prison, or worse if he refuses to go.
So while advocates of state welfare and other state power like to claim the moral high ground in the name of “helping the poor”, or “doing good”, or “liberating foreigners”, they do not actually hold the moral high ground. They are relying on the threat of violence in order to achieve their objectives.
Therefore, it is only libertarians, who oppose the initiation of force for political or social change, who can claim the moral high ground.
Fortunately, the libertarian position also yields the greatest peace and prosperity. A free market will allocate resources in accordance with consumer demand. It is only a system of property rights and free association that will yield the greatest prosperity. Any state intervention that goes beyond the protection of property rights and the enforcement of contracts will make our living standards lower than they otherwise would have been.
Of course, in a libertarian society, people are free to be as charitable as they would like. There is nothing prohibiting the wealthy neighbor from donating money to help the sick neighbor. There is also nothing wrong with you asking your wealthy neighbor for a donation, as long as you are not threatening force and you are not encroaching on his property without permission.
Most hardcore libertarians who abide by a set of principles (mainly, some form of the non-aggression principle) understand both the utilitarian and the moral arguments for liberty. They are both important.
It is rare to find a principled libertarian who abides or fully understands only one of these things.
In other words, if you find a self-proclaimed libertarian who does not fully understand or appreciate either one of these two aspects, then you will likely find someone who does not fully subscribe to libertarianism.
Ludwig von Mises was a utilitarian. He is essentially the face of Austrian school (free market) economics. He saw economics as value-free. He did not present a moral case. He is a rare exception of someone who was a utilitarian only, yet he remained mostly principled in favor of liberty. This is because he truly understood economics, and he understood that free market economics was by far the best system for human flourishing. Mises was one of the greatest economist ever in terms of his understanding. (I would argue there are others who were/ are better at communicating it to the layman.)
Meanwhile, anyone who fully appreciates the fact that libertarians hold the moral high ground because they only rely on persuasion instead of violence for promoting their goals, is apt to remain principled. It doesn’t matter the possible consequences of certain policies; it is not an excuse to initiate force.
You can explain the moral arguments to non-libertarians, and they may even agree with you initially. But then you will inevitably hear something to the effect of, “But we still have to help the poor.” In other words, they still don’t fully buy in. They don’t fully understand free market economics. And they are willing to allow the state to initiate violence on their behalf for certain, maybe even limited, things.
Ultimately, both moral and utilitarian arguments are important in convincing non-libertarians to become libertarian (or at least more libertarian). It is very difficult to win someone over with just one portion. You might change their positions on particular issues with utilitarian arguments, but they will not have a full libertarian view of everything. They won’t apply some form of the non-aggression principle to everything they are confronted with.
The moral case is a critical component in explaining libertarianism. At the same time, it is important to understand economics to reinforce the moral position and to let others know that libertarianism is both moral and pragmatic.