In The Crucible, what arguments for and against the persecution of witches are made in the court?
The arguments that are the in defense of the persecution of witches come from a select group of characters -- Hathorne, Danforth, Parris, Putnam, and the afflicted girls. They have personal gain invested in the accusations. Their arguments include:
-- I you don't know what a witch is, how do you know you are not one?
-- The judges have "seen", firsthand, girls being "stabbed" and bewitched in the court room.
-- Abigail, and then the other girls, "see" Mary Warren, in the form of an invisible yellow bird, ready to attack them.
-- The girls claim, and the judges attest to the fact that, they have fainted and their skin has turned icy cold because of the bewitching spirits in the courtroom.
-- Ruth Putnam claims that George Jacobs' spirit comes to her room at night, to bewitch her.
The arguments against witches are brought about by those who are accused or are related to the accused. They are Francis Nurse, John Proctor, and Giles Corey. They claimed:
-- they have proof that the girls are lying.
-- Corey overheard some say that Putnam forced his daughter to lie so he could get Jacobs' land.
-- Proctor was told by Abigail that the girls are lying.
-- Francis brings to court a list off 91 people who will attest to the goodness of their wives.
As the above answer shows, the reasons given in court only partially explain why the people of Salem felt that persecuting the witches was necessary (or preferable to other courses of action). A complete explanation for this attitude would have to include the unstated motives associated with Abigail and the Putnams.
Abigail's accusations against Elizabeth Proctor are almost certainly driven by a will to take revenge on Elizabeth and by her continued romantic feelings for John Proctor. Abigail never admits this in court, but instead suggests/implies that if evil spirits are allowed to be sent out against girls like her Salem will become a terrible and terrifying place.
The Putnams are understood to have ambitions to wrest land from some of the people they accuse and so want to see those people eliminated so that, presumably, they will not retaliate and/or level charges against the Putnam's later for attempting to defraud the public and steal land under false pretenses.
While the town's avowed, public reasons for seeking to persecute those accused of witchcraft relate to notions of safety from devilry and the like, these darker motives are certainly also present in the mind's of certain citizens.