Excellent question. Of course, the staff is first introduced as belonging to the shadowly gentleman that first meets Goodman Brown as he ventures into the woods at the beginning of the story. This gentleman is obviously meant to stand for the Devil, and his "remarkable" staff clearly indicates the way that this is suggested, coupled with the way in which this man constantly urges Goodman Brown to travel deeper into the woods with him. Note how the staff is first described:
But the only thing about him that could be fixed upon as remarkable was his staff, which bore the likeness of a great black snake, so curiously wrought that it might almost be seen to twist and wriggle itself like a living servant. This, of course, must have been an ocular deception assisted by the uncertain light.
However, what is interesting is the way that this "staff" is carved as a snake, which is of course a symbol of temptation and evil. Note the way that this man constantly offers Goodman Brown his staff to support him on the journey. Also, note what happens when this man gives Goody Cloyse his staff:
So saying, he threw it down at her feet, where, perhaps, it assumed life, being one of the rods which its owner had formerly lent to the Egyptian magi.
The link of this staff to the ones that the Egyptian magi used in their contest of power against Moses in the book of Exodus in the Bible clearly indicates the way that the staff is associated with evil. Let us just consider briefly what we use a staff for. A staff is used to support us, to give us strength and to aid us, making journeys easier. Clearly, if we put this together with the presentation of the staff in this story, it is strongly suggested that the staff therefore represents the act of relying on evil to support us in our lives.