I do not think Shakespeare was racist, and here are a couple of important reasons why.
When Shakespeare created his hero Othello, he made him a warrior, a career of dignity. A Moor, seen in contemporary terms, was someone who was little more than an animal. Shakespeare sensitively portrayed Othello to be a mixture of qualities, just like every human being. This is the opposite of engaging in stereotypical thinking. Rather than depicting a Moor in a stereotypical way, Shakespeare moved beyond the imagination of his contemporaries, and was three-dimensional in how he offered us a look at a Moor illustrated by a genius.
Perhaps the most telling evidence for Shakespeare not being racist is his depiction of Shylock. Many writers have portrayed Jews as being money mad and willing to take advantage of others in business, which is of course a racist stereotype. There are some Jews who are like this, to be sure, just as there are some African Americans, Canadians, Asians, etc., who bear these characteristics. There are examples of every vice and every vulture in every person in every culture, and Shakespeare is obviously aware of this. Perhaps one of the most classic lines in all of fiction is Shylock’s plaintive cry of, “When you prick us, do we not bleed?” This is from “The Merchant of Venice” and makes the point that no matter what ethnic origin a person has, they bleed when pricked with a needle. They cry when they are sad. They are happy when they are tickled. This is brilliant writing at its finest, and is one of the clearest reasons why people should read Shakespeare. His mastery of the craft of writing is absolute. Although he borrowed plots, his use of language transcends anyone else’s in the English language.
He was not a racist. However, he was a genius. I encourage you to find out why if you are already not familiar with his work.