To determine who is the author of the works attributed to William Shakespeare, one would need to consult all the available evidence and then try to make a rational judgment based on probabilities.
Serious academics who have examined all the available evidence have come to the conclusion that William Shakespeare is the author of the works attributed to William Shakespeare. There is actually quite a bit of contemporary evidence to support this claim, and more of it turns up from time to time.
In fact, there is so much evidence (which is easily available; see for instance the literally massive works of Samuel Schoenbaum) that in order to argue against Shakespeare's authorship, skeptics have had to deny or explain away or torturously interpret all kinds of hard data.
If there is a major academic who doubts Shakespeare's authorship, I'm not aware of that person. This is a highly unusual situation, since academics disagree passionately all the time about almost everything -- except this. Skeptics about Shakespeare's authorship often have to imagine conspiracies in the 16th and 17th centuries and then imagine conspiracies in the modern period as well.
Here's a good site that deals with the issue in some detail:
The burden of proof really rests with the people who doubt Shakespeare's authorship. If we had even a tenth of the hard, obvious, undeniable evidence in favor of another candidate that we have in favor of Shakespeare, the debate would be much more lively than it is. The debate, instead, now consists of people making claims that seem dubious.
All I can do is urge anyone who doubts Shakespeare's authorship to examine the claims of the skeptics very skeptically.
One of the most powerful tributes to Shakespeare by a contemporary was written by Ben Jonson, his friendly rival (see link below), who memorably proclaimed,
. . . the race
Of Shakspeare's mind and manners brightly shines
In his well turned and true filed lines
In fact, a very strong case for Shakespeare's authorship could be built simply from the evidence that survives from Jonson alone (unless, of course, he was part of some massive secret conspiracy).