Since you all live in the same home, either one of you can claim the baby as a dependent. You can decide between you who claims her, as long as you both agree. It doesn't matter who supports the baby, as long as she doesn't provide more than half of her own support. The one who is not claiming the baby does not enter any information about the baby on his or her tax return.
Who pays more than half the cost of keeping up the household determines which one of you can file as head of household. The other has to file as single (assuming that neither of you is legally married to someone else). The one who files as head of household has to claim one child as a dependent, but it doesn't matter whether it's the baby or the older child.
