In The Crucible, Danforth represents a "popular mode of power". He makes many of his most important decisions, not based on justice, but based on public perception.
Danforth is given testimony from Proctor that clearly demonstrates the falseness of the witchcraft claims made against many of the people of Salem. Reverend Hale also begs and argues for the release of the accused, saying that the trials have been a fraud.
The popularity of the belief in the idea that the witchcraft is real makes Danforth hesitate to reverse his decision. He is unwilling to go against the majority (despite the prominence of his position and the authority invested in him to act justly). He fears losing face and suffering a dimunition of stature.
In addition to these concerns, Danforth argues that the court has already set a course and a precedent. Some people have already been put to death as a result of the trials.
He argues that it would reflect badly on the court if he released prisoners after executing a number of people accused of the same crimes—regardless of their innocence.
To maintain the stature of the court and to protect his own reputation, Danforth is unwilling to publically recognize the reality of the situation in Salem.