How does one measure it? I’ve already responded to something similar: Nick Nicholas’ answer to Does the Greek language have a variety of regional dialects? and Nick Nicholas’ answer to Which of the Greek dialects sound harsh to a standard Greek speaker?
The most deviant “dialect” of Greek, Tsakonian, is not mutually intelligible with Greek, and outside of Greece is generally referred to as a distinct language.
I’d rather answer the question details:
Which one is regarded as being the closest to the standard language?
Peloponnesian. In fact, as a result of that, Peloponnesian dialects have been studied only minimally; people assumed there was nothing interesting there. Nikos Pantelidis has made his career as a linguist from pointing out that isn’t true; but I fear Pantelidis came along a century too late to find the really interesting stuff.
Which one is considered the most divergent?
Tsakonian and Cappadocian of the obscure dialects. Of the widely known and still spoken dialects, Pontic, followed by Cypriot.
Is there any kind of prejudice attached to those who speak any of these dialects?
Oh yes. They all bear the stigma of country bumpkinness; Greece is culturally very centralising. Northern Greek dialects (which sound the most different, because they’re missing half their vowels) get the stigma routinely; but all the dialects suffer it, ultimately. My cool aunt in Athens told me how lovely and singsong Cypriot was, not like her native hillbilly Thessalian (“stinks of the barn” is how she put it). A few days later, we’d channel-surfed past Cypriot TV, and she said “I always find it difficult to take them seriously, speaking their dialect on TV.”
The nice thing is when the prestige accent of Athens gets counterstigmatised. Mostly in Cyprus, for sociocultural reasons (they’re an acrolect of a live diglossia), but I’m pretty sure I heard the rapid-fire unaffricated speech of Athens mocked in Crete too.