Going back to Kath's post #3:
"The Pier at Dover is under repair; this has long wanted it, not only on the principle of real utility but to give foreigners on their first landing in this island a proper idea of our wealth and greatness. A first impression operates materially. Nothing should be omitted which can by any means make it."
(Kentish Gazette 12 - 15 August 1788, page 2, under "Canterbury" column)
This was of course a very long time before construction of the Admiralty Pier was commenced, let alone the Prince of Wales. The docks were contained within the old North and South pierheads and the landing pier for the sailing sloops crossing from France was inside the tidal harbour.
This painting by William Westall circa 1830 is probaby little changed from 1788. I imagine that the landing pier is that in the immediate foreground with passengers walking up it together with porters bringing their luggage. If the curious structure in the background is the departure lounge then it does indeed need a lick of paint! (joking)
If the sailing craft missed the tide and there was not enough water to enter the tidal harbour, then they would have to anchor off the beach and the passengers would be ferried to the beach by the local Dover watermen. There are a legion of stories about the outrageous charges they would make for their services. The cutter they have landed from is seen in the background.
I think the curious structure mentioned above is just a derrick for offloading baggage and cargo, with bracing to the building behind, although I stand to be corrected. Here is a picture from a later date, after the advent of steam, showing a distinguished French visitor embarking at Dover in 1846. The derrick on the quay is of similar construction.