Bafsk
Bafsk is crawling with imperial soldiers, but unlike those in Fynn, these will not attack you if you speak to them. You'll learn that the Dark Knight left Captain Borghen in charge, and the imperials aren't happy about it. The townspeople aren't happy about the occupation, either. You'll learn that you need to procure a Pass to board the [Dreadnought], and that there is a rebel spy among the imperials. The spy is the soldier blocking a path between buildings in the southwest part of town. Use the [Wild Rose] and [Dreadnought] Keywords, and he will let you continue into the Bafsk Cave.
Before heading underground, take a moment to shop. The Armor Shop has the fantastic Silver Cuirass in stock. This is the best body Armor currently available, equivalent to Mythril Armor but for less than half the price and with a much smaller Evasion penalty. Mages can use it too, and take advantage of its +5% Magic Defense. The only new Weapon available here is the Spear, an upgraded version of the Javelin, but one which is outclassed by the already-available Mythril Spear.
The Spell Shop is selling four new and useful white magic Spells for 800 gil each. Esuna is the main prize, and you'll want to pick up a copy to use after Minwu leaves the party. Basuna is less vital but has its uses if you're willing to level it up. Silence can shut down dangerous enemy mages, though the effect may wear off at any time. Finally, Fear can be made into an effective tool to get rid of enemies, and you don't miss out on any experience points since FFII doesn't use them! You will also miss gil and item drops, though.
Bafsk Cave B1
When you're ready, enter the Bafsk Cave via the staircase in the southwest corner of town. You'll begin in the southeast corner of B1. The area is populated by Monsters similar to those you fought in the lower levels of the Semitt Falls, so you should know how to deal with them. This is a very short dungeon, consisting of only two floors, so it's a good time to go crazy using MP. There's an inn right in Bafsk if you need to recover.
If you follow the water past the first intersection, you'll find a chest containing a Longsword. Take the dry side path instead to continue through the dungeon. The path will soon start following water again. When the water branches, continue south. you'll end up following the water west around a series of walls. The narrow path south leads to a chest with a Longbow, as well as the stairs to Bafsk Cave B2.
Bafsk Cave B2
The enemies on B2 are more of the same, with one new Monster appearing in the rarest Encounter. This is the Dead Head, a monster that is statistically similar to a Soldier (despite their wildly different appearances), but with Undead traits. This indicates weaknesses to Fire, Cure, and Garlic. They should pose no significant threat at this point.
B2 is a narrow and windy floor, but very simple to navigate. There are three ways to go from the start: west, east, and north. Only the east passage leads anywhere, and it will eventually lead to the Dark Knight himself. He's standing in front of a door that opens back on the World Map in the vicinity of the [Dreadnought]. After speaking to him, re-enter the Cave and follow the path leading east. You'll find a room with a chest containing the Pass. There is also a teleporter here that will take you outside of Bafsk. You'll notice if you try to use your own Teleport Spell, it will be blocked, so this is your quickest route out of here.
Back to Altair
Once you've investigated the [Dreadnought] and have the Pass, your work here is done. The imperials have left Bafsk and reportedly headed towards Poft. Purchase anything you still want, and then start the long trek back towards Altair. When you check in on Poft, you'll learn that the [Dreadnought] attacked the towns after it left Bafsk. If you ask Cid about the [Airship], he'll teach you the [Sunfire] Keyword, which you'll be able to make use of in the near future.
Things are much the same in Paloom as they were in Poft. Much of the town's population was killed when the [Dreadnought] attacked. It seems likely it was headed for Altair…