The quest is given by Daisy after talking to her. She will offer the player 200 caps for clearing out the Library, although they can haggle it up to 400 caps (Easy, Medium, and Hard Persuasion challenge, respectively).
To complete the quest, the player has to clear the super mutants out from the Library interior and return Daisy's overdue book at the book return terminal (see article for details).
Afterwards, the quest is completed by returning to Daisy and collecting the agreed-upon fee.
Go to Boston Public Library
Return the Overdue Book
Daisy in Goodneighbor has asked me to clear out the Super Mutants that are threatening the old Boston Public Library. She also wants me to return her overdue book.
Explore the Library
I've returned Daisy's overdue library book. Now to finish the rest of the job by clearing the area of Super Mutants.
Clear the Library of Super Mutants
Return to Daisy
Boston PublickQuoted verbatim, error appeared in the original source Library is safe from Super Mutants. Time to go back to Daisy in Goodneighbor for my reward.