There are a number of ways to insure babies and children. Let us start at the beginning, pregnancy and conception.
In New Zealand currently, assistance with conception is limited to what is publicly funded. Private medical insurance does not cover conception issues; it can cover some pregnancy issues though.
What can you have covered once you are pregnant?
The public system is well equipped for dealing with pregnancy. It is unlikely you will need additional assistance that requires funding if you have a normal pregnancy.
If you do have complications with your pregnancy, then there are provisions under most medical insurance policies specialists and tests benefits to assist with private specialist costs. What you do need to consider are the finer points of the policy, this is where your adviser comes in.