As a libertarian, I don't believe there is anything inherently wrong with the state (or rather any body of government) funding public institutions. Libertarianism at it's core is about decreasing coercion and maxizing liberty. If the people willingly accept the building and funding of such institutions so be it. And if we can reduce education initiatives to smaller levels of government and ensure the same levels of success we should certainly explore these options.
With regards to the structure of the United States however, I would argue that the best method of funding/implementing education (I'm going to include both grade school and college here) is to let the states handle it with the federal government making recommendations (national science foundation funding/recommendations for example). Grade school is a mess because, quite frankly, local school boards are completely useless and not subject to the same level of scrutiny as that of a local city council. Moving these responsibilities entirely to the state would eliminate local boundaries and make local tax initiatives useless. That would also help eliminate the disparities in district funding and influence demand for higher end districts. It's ridiculous that you can have one district with all the wealth right next to a completely shitty district entirely because one side has more corporate taxes or higher incomes (local/state government plays an important role in these problems as well - particulary boundaries, lack of economic development initiatives, and other notable investments.)
I can't speak much on public universities because I don't quite understand fully the funding methods. I do find it interesting however that publically funded institutions charge students for an education but students need loans from the government at interest to pay for it. I would think the states are doing the right thing by funding these institutions, just need to nail down a better medium to pay for it.