Since point $ A $ is constrained to a rectangular region with sides parallel to the axes, its $ x $ and $ y $ coordinates can be chosen independently of one another. The same is true of point $ B $. Therefore, the horizontal separation between $ A $ and $ B $ should be minimized and the vertical separation maximized. The greatest possible $ y $-coordinate for $ B $ is 3 and the least possible $ y $-coordinate for $ A $ is 0. The greatest possible $ x $-coordinate for $ A $ is 2 and the least possible $ x $-coordinate for $ B $ is 4. Therefore, the slope between $ A $ and $ B $ is maximized when $ A $ has coordinates (2,0) and $ B $ has coordinates (4,3). The maximum slope is $ \boxed{\frac{3}{2}} $.