The national debt consists of bonds outstanding, and bonds are held by banks, retirement funds, the central banks of our trading partners, and various rich guys. To "pay down" the national debt, you need to extinguish some of those bonds, or simply replace them with reserves (instead of holding a bond, you would instead hold money in a bank, backed with reserves).
The only way to extinguish net government liabilities (which is what I think most people believe they are doing when they say "pay down the debt") is to run a tax surplus. And unless we are going to attempt to tax China on all the U.S. bonds they hold, or enact a wealth tax on super-rich Americans, the only way to run a tax surplus is to extract dollars from the active economy - the money we earn as income, and spend in the stores, which in turn gets invested to create more production, which is sold to create our income, etc.
Presently, the government adds about $1 trillion in deficit spending to the economy. If the government instead tries to run a tax surplus, you can subtract that $1 trillion right off the top of GDP, then subtract the tax surplus from GDP, and see what lopping $1.5 trillion off of aggregate demand does to the economy. (And that's not even attempting to calculate the knock-on effects of secondary and tertiary spending.) Don't expect tax receipts to remain anywhere near where they are now.