When used as a plea for pity, this appeal to emotion can constitute a potential logical fallacy, while when used as an appeal for sympathy for weaker members of society, or the social good of the long-term health and viability of a society, it can constitute an argument for social justice generally accepted as appropriate.
Claiming to do something for the benefit of children is not a fallacy of itself, but if used to avoid logical debate, it is a thought-terminating cliché. Ethicist Jack Marshall described "think of the children!" as a "tried-and-true debate-stopper" used by "misty-eyed crusaders" to promote government policies and societal actions intended to alleviate the suffering or promote the well-being of children of improvident, impoverished, or otherwise flawed parents, which harms society by removing an important incentive for prudent behavior and by encouraging non-eugenic breeding by the type of person who has "fathered more kids than he can possibly support":
Wikipedia: "Think of the children"
"Think of the children!" is a tried-and-true debate-stopper, but more often than not one that succeeds because of its ability to inhibit rational thought. Children routinely have to suffer the consequences of adult incompetence, recklessness, stupidity, dishonesty and irresponsibility, and if preventing that biologically-dictated result is humanity's priority, then society needs to abolish the enforcement of laws, the obligation to support one's own family, and common sense. [...] [U]nless society sticks to principles that require adults to be responsible regarding the welfare of children in their charge, the "Think of the children!" reflex will suffocate order and justice.
Jack Marshall, "Think of the Children!": An Ethics Fallacy
I don't have access to it, but this may be interesting (and perhaps more scholarly) reading: Paul M. Pietroski, "Think of the Children", Australasian Journal of Philosophy, Volume 86, Issue 4, 2008.