To rephrase unambiguously, change the order of the items in the list, giving
The state and local governments must take into account family types and characteristics of family members in coming up with the policies.
Of course, this only works because the list has only two items and "characteristics of" applies only to one of them. For more complex constructions, such as family types, characteristics of family members and characteristics of family pets, a more significant rephrasing would be needed; perhaps something like
... must take into account family types and characteristics of both family members and pets...
The following suggestion by the asker in a comment doesn't work:
The state and local governments must take into account characteristics of family members, as well as family types, in coming up with the policies.
This is still ambiguous because it could reasonably mean "... characteristics of family members, as well as characteristics of family types..."
An alternative solution is just to ignore the "problem". Anything you might consider about a thing has to be a characteristic of that thing, so "considering family types" and "considering characteristics of family types" amounts to the same thing. This is in contrast to, for example, "I don't want to go to India because of my fear of tigers and the cost of plane tickets", which you might want to rephrase to avoid the suggestion that you're afraid of the cost of plane tickets.