The well-known linguist John McWhorter argues that Shakespeare's plays should be translated into modern English since they are written in a language that we effectively no longer speak. "Indeed, the irony today is that the Russians, the French and other people in foreign countries possess Shakespeare to a much greater extent than we do, for the simple reason that unlike us, they get to enjoy Shakespeare in the langua…
The difficulty with Shakespeare isn't that it's in Elizabethan English, but that it's in poetry. The Russians and French have strong poetic cultures. They enjoy reading poetry, and the challenge of writing it, including translation. That just doesn't exist in Anglophone cultures.
But Shakespeare wasn't meant to be read. It was meant to be performed, and a native speaker watching a good performance has no need of translation. The emotions and flow of the story are clear. They may not understand each and every word, but that's often true of even the latest movies. (The dialogue is often mixed in a way that makes it hard to understand -- by the director's choice.)
I regularly perform Shakespeare for audiences who have never seen it before, and I never translate a word of it. I generally edit it to make it move more briskly than Elizabethan audiences wanted, but the only time I'll change a word is to make the pronouns match the actors I cast (and not even always then).
Shakespeare's stories are rarely worth preserving for their own sake. The stories themselves are dated -- not the words, but the culture. There's no "translation" that will make Claudio not an asshole for the way he treats Hero, or make Hero not look pitiful for taking him back.
Shakespeare's plays in modern words are always flat, dull, and penitential. If you're going to modernize the language, modernize the whole thing -- go see the delightful "Ten Things I Hate About You" or the magical "West Side Story".