The Fed doesn't print money. It does affect the size of the money supply, but it does this with interest rates, securities, and the reserve rate (or can do this). The reserve rate is generally not touched (it causes shocks in the money supply). The last two generally affect lending and saving rather heavily (make the interest rate high enough, it's not worth it to invest; low enough, not worth it to save).

The fact that the dollar is fiat has little to do with the problems we ran into (printed money is only a tiny fraction of the money supply).