Why is it in Europe's interest to halt the drop in the value of the Dollar? Or am I misunderstanding this entirely?
Having a strong currency is great when you're essentially a bank/trading country and want to attract investment funds.
But when you produce anything that you should sell to other markets, it's not, whether you sell services or goods. And it's not good when you're a touristic destination either.
So, currency-wise, you need to strike the right value for a just balance between your financial markets, imports and exports.
The US has been a very strong market for European goods. Fashion, cars, wine, and other food items sell very, very well over there. The US is a prime market for Mercedes and BMW, for example, and for the entire Italian fashion industry.
Hence for the EU, seeing the Euro rise endlessly against the US dollar is not too good.