The fact that the dollar is not going down much relative to the EURO may just mean that the traders are not impressed with the dollar either. The financial storm, when it comes, will be global.
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.