Given that most libertarians would say that they think it right that the state intervene as little as possible in family life, it may therefore follow that it is right to let people who beat their children with or without implements to just get on with it. Their family, their decision.
But hang on a second. From Wikipedia:
"Libertarians support an expansive view of liberty as the proper basis for organizing civil society. They generally define liberty as the freedom to do whatever one wishes up to the point that one's behavior begins to interfere with another's person or property. At the point of interference, each party would become subject to certain principled rules for adjudicating disputes, generally accepting that one who has demonstrated a proven lack of respect for the rights of others should be subject to sanctions, including possible constraints on their freedom. They believe that liberty is the right of every individual, with some viewing it as a natural right."
Yep...there it is. Liberty is the right of every individual. So in the above case, either we regard children as individuals, as proper members of the human race, or we do not. If we do not, then we are right to let families carry on beating their kids. If we do, we then have to decide whether beating a kid is sufficient to the description of someone "beginning to interfere with another person".
Clearly the only answer here is to ask the kid. Sometimes there will be cases where being beaten by the parent does not constitute infringement in the mind of the child, though one would query what inferior theories have been fed to the child in order to allow him to draw such a conclusion. (Of course, the provision of poor theories could well be considered a matter of abuse and constitute a matter of serious interference with another person. But to stick to the main point, for the moment: )
In the situation where the child expresses his view that the action of his parents does constitute negative interference, what should an outsider do? Well as above they should consider the fact that "each party (ie: parent and kid) would become subject to certain principled rules for adjudicating disputes". In other words, proportionate action on the part of members of civil society seems to be appropriate here: one might start with an attempt at re-education, with explanations, arguments; private then more public expressions of disapproval and obvious refusal to liase with those who partake of such behaviour may extend the pressure. But if all else fails, it rather looks as if, should we want to take libertarian principles seriously, the parents who physically assault their children should be subject to the same laws that adults who assault each other should.
How can we raise a generation of freedom-loving kids, if we don't explain and demonstrate that the principles work in as wide a domain as possible? I personally vividly remember noting all those internal contradictions in the religious faith that was handed down to me as a child. It made me sceptical and then a non-believer. Children may well notice if libertarian principles are applied to adults but for no obvious reason, are not applied to them. This may well make them sceptical. And they would be right to be so, not least because libertarian principles can and do work for kids. See eg: Taking Children Seriously, for more explanations.
Children can enact libertarian ideas and should be in receipt of them. They can be rational beyond the expectation of many in the situation that adults give them this chance and bother to explain things in a rational and consistent manner. There is absolutely NO reason or excuse to beat a kid in any rational scheme of things and in the situation that a child does not want to be interferred with in this way, the beater should be subject to the usual laws of assault.