If anything, the dollar has benefitted from EU problems due to so-called flight to safety (odd as that sounds, we’re bad off but they’re worse at least for the moment), and so the decline of the Euro vis-a-vis the dollar would therefore be exaggerated.
Note that the single EU currency was introduced at par with the dollar in 1999.
In an unintended consequence, this could end up propping up the US economy, of all things! And it could tempt companies like Goldman Sachs to look at buying European assets at fire sale prices, if they don't beat the Chinese to it first!