What is Danforth's basic attitude toward the proceedings of the trials, and how does his philosophy influence the trials?

In my opinion, Danforth's basic attitude is that he cares more about how the court and government will end up looking than about anything else.  He does not really care about justice or about witches, he cares about making himself look good.

This influences the trials because it influences him to make decisions based only on the image of the court.  So he ignores the fact that the girls seem to be lying.  Once he has believed them, he has to keep on believing them so that he does not end up looking bad.