Of course, when we think about fears in this excellent short story, the biggest fear that Young Goodman Brown faces is the way that his experience or dream or vision presents his youthful and innocent wife Faith as being involved with the Devil in the same way that all the other good and holy people of his village are involved with him. However, what will be of interest to you is how Goodman Brown reacts to facing his fear. The story is very careful to maintain the ambiguity of what Goodman Brown sees in the woods, and we are unsure whether it is a dream or not. However, what is clear is the way in which Goodman Brown is transformed by his experiences in the woods for the rest of his life, and not for the better. He seems unable to face and accept his fear of the way his dream shows the inherent evil that is present in everyone, even his wife, Faith. Note what the text says about his transformation in the final paragraph of this story:
A stern, a sad, a darkly meditative, a distrustful, if not a desperate man did he become from the night of that fearful dream... And when he had lived long, and was brone to his grave a hoary corpse, followed by Faith, an aged woman, and children and grandchildren, a goodly procession, besides beighours not a few, they carved no hopeful verse upon his tombstone, for his dying hour was gloom.
So, whilst Goodman Brown is shown to witness his fear, the way in which he faces it and stands up to it has to be questioned. We might conclude that he lets this fear dominate him and the rest of his life, sucking all joy out of his existence and transforming him into a serious, grim and joyless individual.