Many minarchists, most notably Ayn Rand, believe their minimal state should be funded by voluntary taxation.  As Rand puts it:

In a fully free society, taxation–or, to be exact, payment for governmental
services–would be voluntary. Since the proper services of a government–the
police, the armed forces, the law courts–are demonstrably needed by individual
citizens and affect their interests directly, the citizens would (and should)
be willing to pay for such services, as they pay for insurance.

The rationale:

The principle of voluntary government financing rests on the following
premises: that the government is not the owner of the citizens’ income and,
therefore, cannot hold a blank check on that income–that the nature of the
proper governmental services must be constitutionally defined and delimited,
leaving the government no power to enlarge the scope of its services at its own
arbitrary discretion. Consequently, the principle of voluntary government
financing regards the government as the servant, not the ruler, of the
citizens–as an agent who must be paid for his services, not as a benefactor
whose services are gratuitous, who dispenses something for nothing.

Economically, there’s an obvious objection to this version of voluntary taxation.  Namely:

1. If the government sells excludable goods, the free market will almost always offer a better deal. 

2. If the government sells non-excludable goods, people have no incentive to pay.

But there’s also a persuasive rights-based objection to the general idea of voluntary taxation.  Namely:

1. Individuals shouldn’t have to pay the government in order to use what they own.

2. Individuals shouldn’t be allowed to pay the government in order to use what others’ own.

Intuitively: If X doesn’t violate rights, it should be legal; if X does violate rights, it should be illegal.  Either way, a government in search of revenue is out of luck.

Or is it?  Even the staunchest libertarian has to admit that rights violations are occasionally fuzzy.  The smoke from a small campfire violates no one’s rights, but the smoke from a mile-wide campfire usually will.  Dropping an ounce of toxic waste in the ocean violates no one’s rights, but dumping a billion tons of toxic waste in the ocean does. 

All this implies a golden opportunity for the scrupulous minarchist.  While the government has no right to totally ban fires or toxic waste, it has a responsibility to prevent people’s rights from being violated by either.  Sure, it could discharge this responsibility with an outright ban on excessive emissions.  But there’s another approach: Impose a Pigovian tax on excessive emissions – and keep raising the tax until emissions are no longer excessive.  This proverbially kills two birds with one stone – protecting rights and raising the revenue required to protect those rights.

What about the rights of the polluter?  As long as the government only taxes emissions severe enough to constitute a rights violation, the polluter has no legitimate complaint; the government is merely deterring him from doing what he has no right to do in the first place.  What about the rights of the pollutees?  As long as the government taxes emissions down to a level mild enough to not constitute a rights violation, the pollutee has no legitimate complaint either; the government is merely allowing people to exercise their rights to light a little campfire.

Note further that Pigovian taxes give government a revenue source without selling either excludable or non-excludable goods.  Instead of trying to make money by selling stuff, the government makes money by charging people for doing stuff they have no right to do in the first place.

From a minarchist point of view, the main strength of the Pigovian approach is also its main danger: the fuzziness of the rights the government is putatively trying to protect via taxation.  Once government gets its revenue from taxing morally impermissible pollution, it has a strong incentive to move on to taxing morally questionable pollution, and then perhaps expand to morally innocent pollution.  And due to the fuzziness, there will always be debate about whether the Pigovian tax authority is overstating its bounds.  Compared to most public choice problems, though, this seems pretty mild – especially assuming the minimal state stays minimal.

I’m not a minarchist.  But if you are a minarchist, Pigovian taxation should excite you.  A minimal state really can fund itself without begging for donations or politely robbing citizens at gunpoint.  In fact, a truly minimal state could probably run permanent budget surpluses.  What would minarchists do with their burgeoning sovereign wealth fund?  Interesting question.