Wouldn't the act of abstaining from an active role in the education of children be considered a "public good" by libertarians? Or at least a "good"?
Should the State prescribe minimal guidelines for education? If not, then how could the State secure the "individual rights" of the child who is deprived of an education (or worse)?
Yes, and generally, as the libertarian government protects the private good, many a public good would ba a side effect of that. We can generalize from this that the libertarian worldview, despite the peculiarities of its analytical method, does not produce anything radically different from the classic view on what good government is.
Should the State prescribe minimal guidelines for education?
The test is whether, if given a time machine, the grown up child could reasonably sue his parents for not educating him. For example, if a child hasn't been taught to read and write, the parent, absent some improbable excuse, would be judged to have neglected the child and deprived him of an opportunity that otherwise would have been clearly his. On the other hand, if a child hasn't been given piano lessons, the lawsuit would fail because it is not clear that having received the lessons the child would have made a Horowitz.
So here's the rub: the government may see public good in producing star pianists. If you think it is far fetched, recall the Soviet Union with its Olympic athletes and ballet dancers. Thus a government would be tempted to go beyond the simple test of individual harm and impose educational standards that would not have been reasonable for a grown up child to retrospectively demand. Should that happen, the rights of the parents would be violated by the government, with secondary loss accruing to the child.
We conclude that the government may impose minimalist educational standards; but allowing the government to do so would be bad public policy, given the government's appetite for growth beyond its legitimate perimeter.