Paine’s criticism of the Quakers comes in an appendix to Common Sense. It is perhaps more appropriate to say that Paine is criticizing the leaders of the Quakers. He is criticizing them for their lack of support for the Revolutionary War.
To understand this, let us first think about what Paine’s goal in Common Sense was and about who the Quakers were. Paine’s goal in this pamphlet was to whip up support for the American Revolution. He was writing in late 1775, when it was not yet clear if American colonists would support independence. He was strongly in favor of fighting for that goal. The Quakers, meanwhile, were pacifists. Their religion forbade violence for any purpose. Therefore, the Quaker leaders were, at the time that Paine wrote, opposed to fighting for independence.
It is for this reason that Paine criticizes the Quaker leaders. He says that they should not mix religion with politics. He says that he hopes
…that the example which ye have unwisely set, of mingling religion with politics, MAY BE DISAVOWED AND REPROBATED BY EVERY INHABITANT OF AMERICA.
He also says that the Quaker leaders are making a mistake in who they blame for the war. He thinks that they should blame the British because it was the British (Paine argues) who started the violence. Paine says it is wrong to blame the colonists for defending themselves. He says that
It is the violence which is done and threatened to our persons; the destruction of our property by an armed force; the invasion of our country by fire and sword, which conscientiously qualifies the use of arms…
So, Paine is criticizing the Quaker leaders for their failure to support the Revolution. He thinks that they are blaming the Americans for the violence when they should be blaming the British. He thinks that the Quaker leaders should not mix religion and politics.