The baby only needs an Extension if they entered Thailand on a foreign passport. If they were born here they are fine until their parents decide to leave and get them a foreign passport.
Where was your baby born? If in Thailand, he/she is fine and legal, as they will have a Thai birth certificate. When they leave a note is made in their (foreign) passport.
Thanks. One more question, once you're registered for online reporting, is the account registered to you (the alien) and therefore you continue to use it even if you move to a new permanent address, or is it registered just for the residence you apply from, and if you move permanent address you need a new login?