According to the Memory Alpha article on apartments : James T. Kirk's San Francisco apartment was the home of Admiral James T. Kirk in 2285.
It was in a tall skyscraper in San Francisco and featured sweeping views of the city, the bay, Alcatraz, the Golden Gate Bridge and at least two other prominent 23rd century skyscrapers.
It dives a bit further into the backdrop itself:
The backdrop view of San Francisco from Kirk's apartment was previously used in the movie The Towering Inferno (1974).
The backdrop in that movie was used for a 135th floor restaurant overlooking San Francisco.
The metallic sculpture in Kirk's apartment near the fireplace bears a striking resemblance to the "Glass Tower" skyscraper in the movie.
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His apartment is in Russian Hill.
The view from his apartment is looking north towards Marin County from the Russian Hill neighborhood in San Francisco.
According to FilmInAmerica.com , This behind-the-scenes picture shows the translight used on stage to create the view out Captain Kirk's window.
It appears as if the view is from Russian Hill since we can see the Marina and Presidio on the left, the Golden Gate Bridge next, and then Mount Tamalpais in the distance on the right.
The Marin County lights, many of which were probably added by the effects crew, are coming from Fort Baker and Sausalito.
View looking northward toward Richardson Bay.
View looking more north-by-northwest, showing left view from Kirk's apartment.
Here is a picture of two artists creating the backdrop for Kirk's apartment.
A few oddities which you can check by comparing the image above to a Google Map of the region .
Angel Island is missing.
It is between Russian Hill and Marin County.
It should be at the far right of the translight backdrop.
(Unless the building lights at the shoreline are on Angel Island, which would be odd since Angel Island is a state park.)
Lights from several cities are missing.
There are no lights along Richardson Bay even though that currently has the cities of Sausalito, Marin City, Belvedere, Tiburon, and Strawberry.
The Golden Gate Bridge is missing.
It should be visible on the far left side of the translight backdrop.
We know from subsequent movies, TNG episodes, and Deep Space 9 episodes that the iconic bridge survives until it was destroyed by surprise Breen attack during the Dominion War sometime around stardate 52,600.
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I think it's on the Mare Island golf course looking inland.
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I'm not sure there's anything within either The Wrath of Khan or The Search for Spock that explicitly says where Kirk's apartment is.
The existence of both fast atmospheric shuttlecraft and transporters mean that it could actually be anywhere at all!
The simplest explanation, though, is that it is in fact in San Francisco or at least in the Bay Area.
Presuming that he actually has a "day job" as an administrative admiral at Starfleet HQ, or as a teacher at the academy, or both, it would certainly be the most convenient location for him.
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I was wondering the same thing after watching Star Trek: The Wrath of Khan on SyFy last night.
Here are a couple shots of Kirk's apartment on Memory Alpha.
The windows look out onto the Bay, but I don't see the Golden Gate Bridge or Alcatraz.
Does the ridge behind the bay look familiar to anyone?
That could help pinpoint where in the Bay Area Kirk's home is located.
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This would also have been his home in "Wrath of Khan"; because it seems to havethe same view out the window. I assume it is in the SF Bay Area but it kind of looks like it is looking back towards Sausalito and I want to know if I'm right.   However I can imagine that his home in the movies is somewhere other than it was in the original series since he is semi-retired.
Where is Kirk's home on Earth in "Search for Spock"?
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It was in a tall skyscraper in San Francisco with sweeping views of the city, the bay, Alcatraz, the Golden  Gate Bridge and at least two other prominent 23rd century skyscrapers. The view is from between Russian Hill and Marin County, looking towards Richardson Bay, since we can see the Marina and Presidio on the left, the Golden Gate Bridge next, and then Mount Tamalpais in the distance on the right. Kirk’s apartment is in Russian Hill or between Russian Hill and Marin County looking northward towards Richardson Bay and the Marin County lights are coming from Fort Baker and Sausalito. 
Kirk’s apartment is in Russian Hill or between Russian Hill and Marin County looking northward towards Richardson Bay. The Marin County lights are coming from Fort Baker and Sausalito. 