What was the major argument against the Constitution by the Anti-Federalists and what was added to alleviate their fears?

Answer: Anti-Federalists argued that the Constitution gave too much power to the federal government, which would diminish the rights of the states and of individuals. The Bill of Rights was added to the Constitution to alleviate their fears.

Further detail:

The Anti-Federalists had opposed ratification of the US Constitution. The Articles of Confederation, in place prior to the ratification of the Constitution of the United States of America, had granted stronger authority to the states. Patrick Henry and other Anti-Federalists were concerned about too much power winding up in the hands of the federal government and its executive branch, thus allowing a small number of national elites to control the affairs of the USA. They feared this also would diminish the rights and freedoms of individual citizens.

The Bill of Rights, laid out in the first ten amendments to the Constitution, provided some reassurance to Anti-Federalists after the fight over ratification, because these amendments to the Constitution served to guarantee that individuals' rights would be protected under federal law.