The simple rule is: The children's insurance should be the type of insurance the higher earning person has (private or public health insurance).
The more in depth rule is: A child can NOT be covered under the public insurance system if one parent is on private and one on public if all 3 of the following are true:
the two parents are married (or are registered partners)
the income of the privately insured parent is higher than the income of the publicly insured parent
the total monthly income of the privately insured parent is greater than €5362.50