Retrieve banner from Dead Town was a mission in Jak II. After protecting Kor (and The Kid) from the Krimzon Guard he directed Jak and Daxter to Torn, a member of the Underground; a rebel movement. However Torn requested that Jak first proves himself by retrieving the Baron's banner from the Ruined tower in Dead Town before he can join the Underground.
The PlayStation 3 and Vita versions of this mission are related to a trophy named "Dead and Gone", completing this mission will reward you with a bronze trophy.
For now Dead Town only has two enemies; the glub and the znorkle tooth not to forget the muddy water which is harmful to the touch. The glub is pretty standard though the znorkle tooth will counter-attack with its tongue upon being hit once, requiring you to be either out of its range or fast enough to kill it before it strikes.
Simply follow the fairly linear path (avoiding the mud) and eventually climb up a ruined wall. Fight some more glubs and climb higher and further into the ruins until you reach a bridge leading a tall faintly-red tower. The bridge breaks down as you walk over it, forcing you to continue.
Several of the stone platforms around this tower are old and break apart when you jump on top of them, they're a slightly different color gray by which they can be identified. Jump over the platforms and continue in this sidescroller bit until you reach a set of safe steps and find the banner.