A common proof-text of this position is John chapter 20, where Jesus says to Thomas (who doubted Christ had been raised from the dead) “Because you have seen me, you have believed; blessed are those who have not seen and yet have believed.”
You can see how the conservative evangelical position leads to a very opposite view of the value of mystical experiences. Under this view, for God to reveal himself directly and experientially to us would be to diminish the value of our “faith” and rob us of an unspecified blessing. Because God would never do such a thing, they denounce those who profess mystical experiences, either as having been deceived by Satan, or as cynical deceivers themselves practicing on the gullible.