It seems to me that we still need to understand the fact that there are a lot of enemies of the dollar and the United States of America. And it is not necessary to assume that the US government prints counterfeit dollars or not supported by gross domestic product, because I believe that any enemy can do this by throwing bad money into circulation. Based on this government we love the way it will protect the dollar, as well as many countries too, because their economy also depends on the dollar.
Aside the enemies that you said do print bad money, there are many of their citizens also that are into bad stuff like cocaine and the rest, they also print bad money and then circulate it within the country which till now, some of them has still not been caught, if US needed to print more money, they would have just done it legally, but they all know the repercussion of over printing of money and how it can damage the economy completely, so they would rather just borrow than to print.
We could see the high debt that the united states is in presently and how they are trying to come out of it, so I could never believe such a thing that the united states would be printing fake money. They are facing the little challenge they have now because of debt that they own and also due to their little bad economy which they will surely come out of it.