jmc

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 $\boxed{\frac{3}{2}}$.