It is unsure if there was an actual “dock” but at one stage there was a daily seaplane flight from Calais to Dover carrying mail (and the occasional passenger). It was not financially successful and failed. Possibly there was a special area of the harbour for the arriving seaplane.
There was also a seaplane base, with headquarters ashore at East Cliff, during WW1, with the seaplanes attached to The Dover Patrol hunting for enemy submarines.
In WW2 a German “Red Cross” seaplane, attacked off Deal, was brought into Dover harbour but destroyed shortly afterwards by a German shell (or bomb?).