Yes the power company can still shut your electricity off for non payment even when you have a baby or children in the house.
The only exception is if the baby or child needs medical equipment that runs on electricity to keep them alive and healthy.
And for that it would need to be proved to the utility company by your doctor and yourself through paperwork stating that the child or baby or person needs electricity to operate the medical equipment.
But just having a baby or children doesn't prevent the electricity from being shut off for non payment.
You can ask around for help to pay your electricity bill or work with your electric utility company on payment arrangements.
Now in the winter months the electric can't be shut off for non payment but you still must be making some payments at least.