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.