I don't know much about the White Cliffs of Dover, but I can offer some personal observations of the coastline around San Francisco.
The cliffs themselves:
When I see cliffs with nice, clean, well-defined edges, I immediately assume the cliffs are unstable. Oceans typically come with lots of strong winds and lots of sand/dust/grit in the air. These phenomonon act as natural sandblasters to the local geology, turning sharp edges into smooth, worn slopes. If a cliff edge is sharp, that says to me it has not been exposed to the ocean air long enough to get worn down. That says to me that the leading edges of the cliffs collapse so frequently, wind erosion don’t have time to have a significant effect on the edges before they fall and are replaced with new edges.
The cliffs in your photo do not have a bunch of structures at their edges. This is probably because (according to weather.com) the cliffs are eroding at a rate of 8 to 12 inches per year.
Ships docking at the port:
Your ships will have a heck of a time docking at your port. Those unstable cliffs have been shedding boulders and who knows what else into the ocean for centuries. This means the ocean leading up to the cliffs have been changing a lot. There probably won’t be a lot of well-known, safe approaches to the land. Ship captains will have to stay far back from the cliffs. There won’t be many soft beaches for the rowboats to approach. Docks will have to be constantly moved around to accomidate the changing landscape. Remains of abandoned docks will litter the ocean around the beaches, causing more navigation hazards to ships. Your city is in for a LOT of upkeep work.
The city atop the cliffs:
People get upset when their houses fall into the ocean. For example, look at the Esplanade in Pacifica, California. In the 1960s, apartments were built a few hundred feet back from the cliff edge. There was a small strip of land, a community swimming pool, and a road in between the cliff edge and the apartments.
In 2010, the apartments were condemned by the city. One of the back porches attached to the complex fell into the ocean. The pool and road were long gone. Many of the apartment’s residents refused to move out. They sued the city in an attempt to remain in their apartments. They claimed the city had a responsibility to protect the apartments by reinforcing the cliffs. The city had been attempting to reinforce the cliffs for years, moving giant boulders down to the narrow beach below and piling them up against the cliff edge. Boulders used in this manner are called riprap.
Spoiler alert: Riprap doesn’t work. The water seeped through the gaps between boulders and continued eroding the cliff.
The city tore down two of the apartment complexes. We’ll see what happens to the rest of them.
In summary, your city can try anything they like, but the ocean will win this fight. The disaster will be slow-moving and a lot of people will refuse to believe it is happening. In the end, whoever owns the infrastructure immediately around the cliffs will likely experience total financial ruin.
White Cliffs of Dover erosion rates:
California coastal town of Pacifica dealing with erosion:
Pacifica apartments almost falling into the ocean:
More on Pacifica, here are a series of photos from 1972 to 2016 showing the erosion progress:
The fate of the Ocean Shore Railroad, built on the cliffs in the same general area as the aforementioned Pacifica apartments:
And the fate of Devil’s Slide (a former section of California Highway 1), about 10 miles south of the apartments: