Much ink has been spilt over a recent whine by a PhD economist at the Federal Reserve bank of Richmond. The crux of the whine is that he has a PhD, and people who don't are ignorant about economics.
Being in the working music field and holding advanced degrees in the subject, I have seen many sides to this debate, at least in music. Basically it should boil down to what a person does, says - and how they play - not what degree they hold. There is way too much emphasis on degrees as a measure of status in the U.S. at the moment, probably a function of our increasing class-based social stratification.
At any rate, a person with an advanced degree in music is quite used to be challenged by very well-informed amateurs on all fronts, from interpretation to history to what is 'good' and what is 'bad', even up to playing ability in some cases. It seems this is less true for other fields.