How did folklore influence Shakespeare?
Shakespeare was influenced by stories he heard, read or was told, which would be consistent with the definition of Folklore, stories that are passed down through the oral tradition.
"I imagine that going to the theater was a bit like this for Shakespeare's audience. The plays they watched were not original stories, but familiar ones: English history, adaptations of Italian novels and Roman comedies, retellings of ancient British legends. Playgoers in Shakespeare's time attended plays for the pleasure of hearing beloved stories well told. They also got familiar folk songs, the excitement of seeing well-trained bodies in motion, and a few special effects created out of not much more than bits of string."
Evidence of Folklore can be found in several of Shakespeare's plays, including Macbeth, use of the witches, King Lear, he uses the atmosphere of the carnival life to illustrate Lear's chaos, A Midsummer's Night Dream, he uses fairies, which were part of British Folklore tradition.
The attached link is a rich source for how Shakespeare applied Folklore traditions to the writing of his plays.