Zakat al-Fitr should be paid on behalf of everyone in the family. There are some scholars that recommend that Zakat al-Fitr is also paid on behalf of unborn children after the 120th day of pregnancy, but do not view it as obligatory. Most scholars do agree, however, that Zakat al-Fitr should be paid on behalf of the baby after his/her birth. Please do consult with your local imam or scholar for further clarification.