It should be financed by taxes, which includes caveats like eliminating virtually useless degrees like music education, drama, fashion design, anthropology, etc. People who want that can pay out of pocket for private institutions.
If you want to defund the areas of education that don't provide value, you should favour less government intervention, not more. Let the market fund education by having employers or financiers preferentially fund (through loans, or otherwise as discussed in TFA) the students who are more likely to enrich them later on.
A lot of education doesn't necessarily produce "value" in short term bean counter sense, but are very important to for example retain culture and knowledge.
Also, to apparently great shock of many, there are a lot of people who don't select what to study based on what is "more likely to enrich then later on". Most people don't care much about being rich, and the ones that do are probably quite miserble.
Even if we decide the only relevant criterium is the free market, people pay for the ability to do these things, and for good reason, just as they pay for the ability to use and develop technologies.
I think it would be a great loss to all of humanity if we suddenly stopped paying for social science education. In a nutshell, and a bit polemic: What do you want humanity to be, a great many cogs in a science machine churning away without looking sideways? I would not want that.
It's nothing but an arbitrary choice that there is an art degree, but not a heavy machinery operation degree, and not the other way around. This is all class based gate keeping, and should never be formalized and funded by tax money.
Seems fairly obvious why three years of sitting in classrooms and libraries isn't seen as a particularly useful way of demonstrating aptitude for operating heavy machinery.
Just because some fields don't (pretend to) have formalized theories, it doesn't mean the study isn't rigorous.
Some fields have highly competitive admissions, because the number of people willing to study them far exceeds the demand for graduates. This may be because the field promises a well-paying high-status job (e.g. law, medicine) or because it is inherently interesting (e.g. social sciences, biology). In other fields, particularly in STEM, admissions can be easy, because the demand exceeds the number of capable and interested students.
The only degrees where this selection mechanism seems to have inverse effects are business management related ones.