The major impact that Shakespeare has had on our society is in the fact that so many people are required to study his work in high school and college. Because we have all read or watched at least some Shakespeare, his works give us some amount of a common cultural background -- references that we all understand, etc.
Just think about all the Shakespeare references that are quite familar to us all. Everyone knows who Romeo and Juliet are -- if you call someone a Romeo everyone knows what that means. I just read a thing referring to a "star-crossed" quarterback -- another Romeo and Juliet reference. We all know about "to be or not to be," "shall I compare thee to a summer's day," and many other such quotes.
Shakespeare gives us something of a common set of cultural references and has, in that way, affected our society.