Hmm. I'm going to go out on a limb here and say that no, the look of multiple piercings is not classy, but neither is it inherently classy to have one hole in each ear, or no holes at all in either ear.
Here's where it gets tricky, though. With one hole or no holes, the class factor is entirely determined by the earring. If you have a large number of holes, to my eye it's always going to be a more juvenile (or, as Rachylou put it so well, "downtown") look.
So by my calculus, one earring in each ear CAN be classy but isn't automatically classy, while multiple ear piercings (I'm going to define that as more than two holes, and/or holes up in the cartilage area instead of in the lobe) will always look a bit more "other", a bit more RATE, a bit more extreme.
Is it harder to look "classy" when you're looking other, RATE, and extreme?
Yep. That's kind of the point, isn't it?