First, the Federal Reserve, which is an independent entity from “the 3 branches” of government controls the printing of money and the supply.

Next, most of the money supply is electronic and not paper. It is money in bank accounts that moves by wire from bank to bank. To manage the money supply there is far more to control than just the printing of money. Their primary tool to manage the money supply are interest rates. Particularly the rate paid to banks to hold reserves at the Fed. The level of this rate and how it compared to other rates (credit card rates, commercial lending rates, mortgage rates, etc.) give these banks incentives (or take away incentives) to underwrite new loans. This creation of credit is the primary mechanism by which money is created.

But, to the point of the question, the Federal reserve doesn’t necessarily want a “strong dollar.” They want a stable dollar. Or even more precisely a stable price level. The Fed also has a mandate to manage economic growth with the goal maximal sustainable employment. Sometimes these two mandates are at odds with one another, and the Fed must decide which is more important at the time.