Children are an important aspect of the world of Frostpunk. They are, in a sense, the source of hope for the city. Because of that each decision related to them has a huge impact on discontent and hope bars. A decision to send them to work depends on your city's situation.
Avoid assigning children to work if your city has enough workers and you are sure that this won't change. Instead, after signing "Child Shelter" law, you can decide who will receive children's help: doctors or engineers. Thanks to that you can raise efficiency in certain buildings without negative consequences.
Let them work if your city suffers from not having enough working force. The first version of this law allows children to work only in certain places - Cookhouse or basic buildings that excavate resources. Here, it is better to order children to work in a Cookhouse. A risk of accident is much lower (child accidents aren't welcome by your society).
As a final resort, you can update the law and let children to work everywhere. After that they become full-fledged workers. This law should only be used in critical situations when your city's survival is in danger. Try to assign children to less demanding buildings and jobs. Adults should work in various mines.