The actress who plays the perpetually depressed Dana, Morgan Saylor, is an incredibly talented young actress. It's not her fault that the material she's given on Homeland isn't all that great. That said, this is about the character of Dana and not the real-life person behind her.
But even then, what is it about Dana that gets people so full of rage? Is it that she tends to burrow her eyebrows, thus wearing her teenage angst on the upper portion of her perfectly adequate face? That she has a nervous tic where she plays with her fingers a lot? Or maybe it's that she attempted suicide after she found out that her father is a freaking terrorist!
Sure, Dana Brody can be a little irritating sometimes, as all teenagers in real life can be. But the hatred toward this one (relatively unimportant) character is wildly disproportionate to whatever "offenses" she has committed in the eyes of the viewing public.