The sentence is definitely wrong as it stands. But I can't say what it should be, because I don't understand what it's trying to say.
You can use between in a few ways:
between right and wrong
between two books
between the lines
But who does what is a singular thing, and between doesn't work in conjunction with that phrase.
Without completely rewriting the sentence, this is the closest I can get in terms of making it grammatical:
Rapid convergence in the media (who) and entertainment (what) industries is blurring the line between "who" and "what."
It's also possible to change the preposition from between to in, and leave everything else the same:
Rapid convergence in the media and entertainment industries is blurring the line in "who does what."
But while that's also technically grammatical, it makes even less sense to me than the first version. (What does who does what refer to, and why is there a line in it that's affected by the convergence of the media and entertainment industries?)
However, it's possible there is more context than the single sentence makes apparent.