The Anti-Federalists had more influence on the Bill of Rights, as they were the ones who wanted it in the first place. The Federalists thought a Bill of Rights was unnecessary, while the Anti-Federalists pushed for it.
The Articles of Confederation was the United States' first constitution. It was very weak and gave little to no power to the federal government. The federal government could not tax the states, regulate trade, create and enforce laws properly, and numerous other important matters. The Anti-Federalists agreed with this, as they associated having a strong government with British tyranny. They thought that if the government was powerful, they may abuse the citizens. The Federalists on the other hand wanted a stronger government. They believed that having a strong government meant having a strong country, and they thought the Articles of Confederation was doing more harm than good. When the Constitutional Convention of 1787 occurred, the Articles of Confederation was only meant to be changed. It ended up getting thrown out all together and the United States Constitution was created instead. The Anti-Federalists disagreed with this and demanded to have a Bill of Rights, which would protect the citizen's rights. The Federalists thought this wasn't needed, as the states had the powers that the federal government did not. The Anti-Federalists insisted though and believed it was important that they had a proper document protecting their rights.
The Anti-Federalists had more influence while creating the Bill of Rights because they were the ones who pushed for it in the first place. They thought having a document to protect the rights of the citizens was needed, while the Federalists thought it was essentially useless as the states already had all of the powers that the federal government did not hold.