"Watchtower" is the key word in our song's title, and it has several connotations. The most glaring connection is to the Book of Isaiah in the Old Testament of the Bible. Isaiah was one of the major Hebrew prophets. Like a lot of prophets, he hated idolatry, or the worship of false images of God. In the quote, a chariot with two horsemen arrives to report that the great pagan empire of Babylon has fallen. Babylon is frequently cited in the Bible as the capital of all things irreligious, especially idolatry.
So here's the quote. It's a long one. Book of Isaiah, Chapter 21, verses 5-9: "Prepare the table, watch in the watchtower, eat, drink: arise ye princes, and prepare the shield./For thus hath the Lord said unto me, Go set a watchman, let him declare what he seeth./And he saw a chariot with a couple of horsemen, a chariot of asses, and a chariot of camels; and he hearkened diligently with such heed./...And, behold, here cometh a chariot of men, with a couple of horsemen. And he answered and said, Babylon is fallen, is fallen, and all the graven images of her gods he hath broken unto the ground."
So, the watchtower is where the soldiers defending the city would be posted to scan the horizon for invaders. It's a protected position, and Dylan likens it in this song to a bastion of privilege. In Isaiah, however, the feared invaders never show up. Instead, a messenger shows up to report that the enemy city of Babylon has fallen. The quote depicts the moment when the defenders learn they may no longer need to defend.
Two other connections are worth noting. First, the word brings to mind a prison watchtower, used to keep people from getting out instead of preventing them from getting in. Every good prison movie usually has a sinister watchtower with a big searchlight that scans the yard. Second, the French also built watchtowers in Vietnam back when they were still the rulers of the roost there. For a song released in 1967, at the height of the Vietnam War, this is not an insignificant detail.