Trade deficits grow when the economy expands and shrink when it shrinks. Thus, we could tank the economy and have the strongest currency in the world. This is a reason the Euro is strong because the continent has unemployment rates about twice ours.
Collapsing the economy is not the way to go.
That's pretty much what Greenspan said on November 19th.
However, the Euro is strong for a couple of reasons. Their interest rates are too high and some countries have started to use it as a reserve currency. An economic boom can cause a trade deficit, but a stagnant economy is rarely the cause of a strong currency.