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.
I assume when Shakespeare wrote his plays, he used the everyday English his audience would have been familiar with. Just because nowadays his language sounds archaic doesn't mean it was intended to be that way. I think a translation into modern-day English would allow us to experience his works the way people in the old days would have experienced it. Of course, purists will disagree, but it's not like the originals aren't going anywhere so I would be in favor of such an effort. Furthermore, I don't think a translation into modern-day English would be so different from translating Shakespeare into a different language entirely.
It's not Shakespeare's fault that many modern people have a painfully poor grasp of their own language. Altering the words of a wordsmith is stupid; it's like repainting the Mona Lisa in a more modern style and hanging the new version in the Louvre.
I like modernized Shakespeare as much as the original works, but...
The problem with modern updates of Shakespeare is that you're very likely to bump Shakespeare up to at least an R rating. It's definitely not PG anymore without the old language to obfuscate (for those who don't understand the antiquated phrasing and references) the innuendo and violence in a lot of his works.
Now, if you're familiar with Romeo and Juliet, you know it features violence, lewd sexual comments (discussion of violent non-consenting sexual acts as well - Sampson is a vile human being), and suicide in its original form. It doesn't take much thought to realize that updating it pushes the work very firmly into NC-17 territory.
I've actually seen a performance of Romeo and Juliet performed in modern English, set in war-torn Bosnia. It was brutal, violent, and terrible, but it made Romeo and Juliet's love for one another all the more poignant for it.
For the average audience member, modernizing Shakespeare probably won't make the work lose any of its oomph. If anything, it probably helps to put it into proper perspective, and have more impact as a result.
Technically, Shakespeare is written in Modern English.
If the language is challenging you, then I recommend relying on a decent annotated edition of whatever play your reading. The difficulty of reading literature from that error is not the grammar, more understanding the vocabulary and some of the context. I also think that hearing it recited or watching it performed helps a lot. My English teachers would always have us read the play out loud in class, then we'd watch a film adaptation, and I must say the combination of both was really helping in terms of understanding what was going on.
The problem with translating Shakespeare is that his plays are absolutely loaded with linguistical nuances and plays on words. In fact I'd say that his ability to play with language is the key allure of his plays. If you changed the language, you'd lose in translation most of what makes Shakespeare so rich and inviting to actors. Anyway, a good classical actor performing Shakespeare can make the antiquated language make perfect sense to modern listeners.
I think that just about anybody with experience in translating will tell you "no," it isn't possible to preserve that level of nuance into modern English. Just trying to recreate the same meaning whilst remaining reader-friendly is difficult enough. Preserving things like metre and puns and rhymes and double entendre when they're all stacked on top of each other the way they are in Shakespeare is simply impossible.