According to general relativity, matter falls down a gravitational field because they are going straight in curved spacetime, not because they are being pulled by a force which is proportional to their mass. So everything should fall down the same in a gravitational field, including antimatter.
Actually testing whether antimatter falls down the same has been rather difficult, given that it annihilates with any matter it touches, but scientists at CERN have finally confirmed that antimatter falls down a gravitational field the same way as matter does, and that it doesn’t fall up.