No. You know it's a real threat when you actually have evidence or suspicion that it might be. If a guy says in the streets "I'll kill you motherfucker" while walking away, just because you accidentally bumped into him, you will not call the police and say that somebody want to kill you. If he comes to your house and say that, then yeah, call the cops and arrest him.
I never said that I like death threats, nobody like death threats. But that does not mean that they are real threats. Just because a random, anonymous person on the internet says something, does not mean it's real.
This remind me of a guy who spent 7 month in jail because he said in League of Legend, as a joke, "Oh yeah I'm real crazy, I will go shoot a school of children, Lol". A canadian women saw it, called the cops and the kid spent 7 month in jail for a sarcastic joke he made while playing video games.
So yeah, death threats sucks, but we didnt care that much when every single public figure out there got them all the time, why would I start caring when it's an another one? If she get real threats, with foundation and some real possibility of harm being done, then I will walk right behind you to denounce the person who did it. Because that's terrorism. Until then, no I will not really care that a public speaker get mean comments about her, when every single person who make any type of content, including me, gets them on a regular or semi regular basis. Talk to any celebrity online, they will tell you they get death threats. Hell, the admin of Neogaf probably get some in his email from angry people all the time. I saw them on youtube pretty often.