On the grounds of style and content this is certainly not by the Buddha.
I don’t know the actual origins at the moment, but it looks like a variant on “What’s done to children, they will do to society,” which is usually attributed to Karl Menninger, an American psychiatrist who founded the Menninger Foundation and the Menninger Clinic in Topeka, Kansas.
Menninger is often cited as having said this, but so far I haven’t been able to confirm that he actually did. The earliest reference I’ve found is in a 1979 book by Barbara Rowe, called “The Book of Quotes.”
I can’t offhand think of anything the Buddha said that relates to the idea that the way we treat our children becomes the way they treat others. But so that you can get a sense of how the scriptures are phrased, here’s part of the Sigalovada Sutta, where the Buddha gives a series of teachings to a householder, Sigalaka, whose spiritual practice involved paying reverence to the six directions (the four cardinal points, plus above and below):
In five ways should a mother and father as the eastern direction be respected by a child: ‘I will support them who supported me; I will do my duty to them; I will maintain the family lineage and tradition; I will be worthy of my inheritance; and I will make donations on behalf of dead ancestors.’
And, the mother and father so respected reciprocate with compassion in five ways: by restraining you from wrongdoing, guiding you towards good actions, training you in a profession, supporting the choice of a suitable spouse, and in due time, handing over the inheritance.
In this way, the eastern direction is protected and made peaceful and secure.