Agreed the USD is the currency of choice for black markets across the world, but it too is subject to inflation. If the dollar loses value, purchase and exchange power is lost. Many smart investor watch the black market to judge true currency strength. So in the end, the idea that USD oversees mitigates inflation does not hold. I would direct you to this informative piece on inflation. The comments by Freepers are very good Link
I just scanned some Federal Reserve documents (like this one, albeit it is eight years old, or this one). They show that foreigners actually hold more cash than in the USA, 55% of the total money stock. Nor are they talking about that money only being used here. BTW, I doubt seriously their analysis of the total cash floating around in the drug trade. They said it is only a $20-30 billion cash stock (95$) based upon the average domestic currency turnover rate of 30-50 times per year! I'd suspect the drug trade to be a third of that turnover rate given the nature of the supply chain. Given that the total amount of currency has tripled since 1995 my guess is that there is a half trillion floating around abroad. That ain't chump change when it comes to the total amount of cash they print.
Another big mitigating force against inflation right now is investment in precious metals, an investment that produces nothing and reduces monetary velocity considerably.