So much of the appeal of Shakespeare's plays is in the rhythm and choice of words that I don't think there's much point trying to modernise it. It would be like translating a Knock Knock joke into Chinese. You can either provide a literal translation that makes no sense--"Knock Knock. Who's there? Potato. Potato who? Potato my left kneecap, my aunt has a terrier called Steve!"--or you can come up with something that works in Chinese, but is basically a different joke.
If people can't grasp Shakespeare, they probably shouldn't be reading it: they should be watching it. In fact, I'd recommend that to anyone not actually writing an essay on the subject (and even those people should also watch the play). These things were intended to be performed, not read. They're better that way. And as an added bonus, it's much easier to understand the language when you can see what's happening on stage.
No one can understand a story in the same way its author did or does. Therefore, any attempt at translation or adaptation merely muddies the original intent and expression of the author. You are editing their voice.
Shakespeare's language isn't really antiquated, though. I've seen a lot of people call it old English, but it isn't. It is actually Early Modern English.
The problem is that it's poetry. It's metaphor and imagery and ideas. He doesn't just say something; he uses words to paint a picture. When people learn Shakespeare in their high school English classes, they're told that it's poetry and then seem to interpret that as needed to add a pause to the end of each line break. The line breaks are only there for the aid of the actors, but it should all be said fluidly.
Once more unto the breach, dear friends, once more, or close up the wall with our English dead. In peace, there's nothing so becomes a man as modest humility: but when the blast of war echoes in our ears, then imitate the action of a tiger; stiffen the sinews, summon the blood, disguise fair nature with hard-favour'd rage; then lend the eye a terrible aspect; let pry through the portage of the head like the brass cannon; let the brow o'erwhelm it as fearfully as doth a galled rock o'erhang and jutty his confounded base, swill'd with the wild and wasteful ocean.
It's poetry that's best read as prose. Nothing antiquated about it, aside from possibly the fact that favour'd and favoured would be pronounced slightly differently (favoured is three syllables, where favour'd is two).
There is absolutely no reason for Shakespeare to be 'translated.' The language is exactly the same as it is now with less-orthodox usage for our day. But as some have mentioned before, annotated editions are quite useful if there is trouble understanding the language. Those kinds of things will help with words and phrases that have lost colloquial value throughout the years.
You've never heard of No Fear Shakespeare? [link] It's pretty useful if you have to read Shakespeare in 9th grade and you can't figure out what the heck anyone is trying to say. I think it has a lot of trouble with the double entendres and other wordplay, since that kind of stuff is always hard to translate.