Surprisingly, little different than before (or even during) the war. Appointment still had to be made by a federal elected official (representative, senator, vice president, or president), so until each state was "reconstructed" and had representation in Congress again, numbers from the South were a little low. That only lasted a few years, and it's worth noting that there was no requirement that a cadet had to be appointed by their own congressmen; thus, many southern men received appointments from northern congressmen, but were still listed as being appointed from their own state.
(A good prewar example of this is George Pickett, who was from - and listed as being appointed from - Virginia, even though his actual nomination came from Congressman John Stuart of Illinois.)
What is bizarre, though, is that during and just after the war, at first glance it appears that there is little drop off in southern appointments. This is, however, misleading, because some of the men appointed during the war and Reconstruction "from" Confederate states were actually residents of northern states whose fathers were serving in the south, and thus the young men were often listed as being appointed from states they personally had never been to. Some of the others had actually served as enlisted men or volunteer officers in the war and were appointed to West Point directly from their volunteer regiments, but sometimes from the states they were serving in at the time and not from the state they actually resided in. Of the 63 members of the class of 1867, for example (men appointed in mid-1863, in other words), 31 are listed as being residents of Confederate states. Of those, however, 17 of them were actually residents of northern states (including the fifth-ranked graduate, Welsh-born Joseph H. Griffith, appointed from North Carolina while serving in Mississippi as a second lieutenant of the 22nd Iowa Volunteers).
After Reconstruction ended and all of those states had full Congressional representation again, everything pretty much returned to normal.