Seriously, folks: this cannot be that hard, especially at Level III.
If you’re concerned about the price of something dropping, you buy puts (at a price below the spot), and to finance that purchase you sell calls (at a price above the spot).
Here, the US fund manager (you) is (are) concerned about the USD appreciating vis-à-vis the EUR, or the EUR depreciating vis-à-vis the USD: he’s (you’re) worried that the price of the EUR (measured in USD) will drop. Your action is clear: buy puts on the EUR and sell calls on the EUR; i.e., buy USD/EUR puts (below the spot) and sell USD/EUR calls (above the spot).
Ah … you say … but that’s the rub: we don’t have USD/EUR puts and calls; we have EUR/USD puts and calls.
The solution is the paradigm of simplicity: a USD/EUR put (below the spot) is an EUR/USD call (above the spot), and a USD/EUR call (above the spot) is an EUR/USD put (below the spot): buy EUR/USD 0.68 calls, and sell EUR/USD 0.64 puts.
Voilà!