I was the person who sorted through letters to the editor for a major daily newspaper. The sheer number of letters we got was incredible, and almost daily they insulted our writers, our editorial board, me personally or the newspaper or journalism in general. Often they included racist, sexist, ageist or just plain prejudiced remarks; sometimes they went so far as telling us we should commit suicide, quit or otherwise do dangerous, shocking and horrible things. We sent them to the circular file as a matter of policy. (Actually, I kept them in a stack and had a policy of not publishing only borderline-offensive letters by frequently prejudiced or threatening writers, but this wasn't universal.)
Responding to idiots (or taking their comments personally) just feeds the beast, even if those particular idiots are articulate enough to be read on-air. I'm disappointed that she gave into this alluring temptation, though in reading the thread I understand the situation better. It's brought a person who depends on popularity among viewers a little more job security, sure, but I'm just not a fan of airing your dirty 'fan' mail on air no matter what. This should've stayed private.
I have one former colleague, a female sportswriter of considerable talent who doesn't look the sportswriter part -- lots of makeup, bright clothes and heels, older than the average female sportswriter and overweight -- and she gets tweeted at or emailed about how ugly/fat she is every time she says something someone doesn't like. It's clear it gets to her, and I feel terrible for her. But at the same time, it'd be SO unprofessional (given our business's code of ethics) to retort or even respond to anything but the sports part of their messages.