Right now, the Federal Reserve is slowly selling off it's $4 trillion in assets, an action which will tend to contract the money supply, slowly raise interest rates, and cause the dollar to appreciate in value.
The European Central Bank, meanwhile, is adding to it's assets, an action which will tend to expand the money supply, suppress interest rates, and cause the Euro to depreciate in value.
Actual currency exchange rates depend on many factors other than these actions, so predictions about relative values are not reliable.
Still, I'd be more comfortable holding savings in dollars than euros at the moment. A bit of both, diversity, might be prudent.
Note that these asset holdings are, macro-economically, equivalent to a sovereign wealth fund. When viewing the debt of the US government, the Federal Reserve holdings should be subtracted from the Treasury debt.
This is true because the income to the Federal Reserve is remitted to the Treasury, after subtracting operating expenses.