You can’t “appeal” but you can talk to a financial aid officer and go over the numbers as to how the aid is calculated, in case there was mistake or something that was not taken into account. So if the amount they offer seems to leave a big gap over and above your FAFSA EFC, you can start by asking what income or assets are the basis for that amount.
But wait until you see the award first – you might be pleasantly surprised. Barnard does meet full need, it is just that their concept of “need” might be different than yours.