Libertarians in favour of a minimal state typically base their case for the state on a public goods argument, e.g.: "Everyone would like to be protected by defense and law enforcement. But it's not possible to make those goods exclusive, so that only those who can pay for them are able to use them. Therefore, they must be provided by a state that taxes everyone for their provision, or there will be too little of them. To keep the resulting state minimal, we need a watchful populace."
But this case crumbles like a house of cards at the slightest touch of a finger to its weak point: Minarchists are asking the populace to solve a much worse public goods problem than the one they started out with. If people cannot work out a solution to the problem of petty criminals' depredations that handles the issue of free riders, then how in the world are they going to solve it when it involves defense against a state to which they have surrendered all of the large weapons, all legal authority, and tremendous resources (from taxation)?
Of course, people can ad do solve these problems, otherwise the 1989 revolutions in Europe would have beeen impossible. And, just so, they can solve the initial problems as well.
As Anthony de Jasay put it, the minimal state is either unnecessary or impossible.