""Then doesn't it follow, as we said to begin with, that the amusements in which our children take part must be better regulated; because once they and the children become disorderly, it becomes impossible to produce serious citizens with a respect for order?"" - Plato Republic, 424e.
Plato argues that the reasonable men in his republic will learn to be so only through a system of education and upbringing that is strictly controlled. This is important to Plato because reasonable men will then understand and accept their position in the Republic. One danger in this equation is the presence of disorder in the education of citizens. This passage refers to a measure of control recommended to fight that disorder. The goal of the educational system is to produce "serious citizens" who understand, agree with and maintain the social order put forth by Plato. This passage is a part of the Republic dealing with the unity of the Guardian class. The details of the Guardian lifestyle are put forth in order to explain the duties of that class. In unity, the Guardians must understand the importance of education in maintaining the Republic. Above all, the Guardians are charged with maintaining the educational system and protecting it from change. Innovation in regards to education is said to be the downfall of the Republic. The disorder described in this passage is thought to be an enormous risk to the balance created in the Republic. If disorder "creeps" into the education system, it would be impossible to create citizens that would fit into and benefit the Republic. Regulation of that system and of children's amusements would keep the children's education in-line, allowing for creation of proper citizens thus perpetuating the Republic.
I believe this passage to be more about the ills that affect all human societies than the regulation of the young and their amusements. This passage describes a method for regulating the young in order to eliminate disorder in society.