The anti-federalists believed that governments would be more responsive to the people and, thus, were less a danger to their liberties.

The anti-federalists believed that STATE governments would be more responsive to the people and, thus, were less a danger to their liberties.
Unlike the federal government that is responsible for the whole nations, the State Government only responsible for the well being of the citizens that lived in their states, so it is easy for them to become more responsive to the locals.