Starbase 1 was composed of a large central spherical habitat and a space dock section connected to the equator of the sphere. The space dock began with an inner ring with six spokes ending in saucer sections. These multideck saucers, labeled from A to F, each contained berths, numbered from 0 to 6. The berths had retractable docking clamps designed to attach to the hulls of starships docked at the station. (Star Trek; Star Trek Into Darkness)
In 2258, the USS Enterprise was docked at the starbase before its maiden voyage, when it left to answer a distress call from the planet Vulcan alongside the USS Antares, USS Armstrong, USS Farragut, USS Hood, USS Mayflower, USS Newton, USS Odyssey, USS Truman, and the USS Wolcott. Seven of these ships, except the Enterprise – which left late, due to an error on Sulu's part – were later destroyed by the Narada during the destruction of Vulcan. The starbase was apparently unharmed while the Narada subsequently assaulted Earth. (Star Trek)
In late February 2259, according to a plan of the base, there were nine starships docked at this base. These starships were given unique two-letter and three-letter IDs. Some of them were CBM, NCC, and ROM. A Space Dock Manifest listed two Constitution-class starships (AF and VH-5), VD, and DB-5. In addition, a news network officially sanctioned by the UFP News broadcast a report about this spacedock. Information on the spacedock and the news report was visible in a collection of graphics and video media on a powerwall in the office of Christopher Pike at Starfleet Headquarters.
In the aftermath of John Harrison's attack on Starfleet Headquarters, the starbase again served as departure point for the Enterprise upon the ship leaving for Qo'noS. As Captain James T. Kirk ordered that all moorings be retracted, the docking clamps were removed, just prior to the Enterprise leaving. (Star Trek Into Darkness)
While this alternate-reality starbase is not referred to by name in the film Star Trek (beyond the lines "dock control" and "separating from spacedock"), the movie's script referred to the facility as "Starbase One". The station was also scripted to be "massive" and appear at the end of the film, though it is absent from the conclusion of the film's theatrical cut. 
The genesis of the starbase's design was the concept of an Earth-orbiting spacedock with a central hub. (Star Trek - The Art of the Film, p. 61) This design was devised by the film's production designer, Scott Chambliss. Concept artist Ryan Church recalled, "Everybody had taken a stab at the spaceport, then Scott came up with that idea; I did a lot of illustrations of variations on the basic concept."  It was Church who came up with the concept of the starbase having a clear, spherical inner core with radiating docking arms. (Star Trek - The Art of the Film, p. 61) Heavily inspired by this design, Bruce Holcomb – a digital model supervisor from Industrial Light & Magic – built the space station as a highly detailed digital model, using CGI. (Cinefex, No. 118, pp. 56 & 61)
According to the Star Trek Encyclopedia (4th ed., vol. 2, M-Z, p. 315-316), this station was referred to as "Spacedock 1" in the script for Star Trek Into Darkness. The authors of this reference work believe the names are interchangeable.
For Star Trek Into Darkness, ILM reused its existing model of the starbase, adding details of operators in the dock's interior as well as the docking clamps, retracting from the Enterprise's hull. (Cinefex, No. 134, p. 82)
Similar to the film's script, the novelization of 2009's Star Trek refers to this alternate-reality space station as "Starbase 1". The book also calls the facility "a city in space" and considers the docking situation depicted in the film – whereby the station's gigantic transverse arms are initially all occupied – as being unusual for the starbase.
A digitally rendered depiction of Starbase 1 was included in the 2011 edition of the Ships of the Line calendar, in the image for June (entitled "His Star Trek Will Go On Forever"). This image, created by Tobias Richter, shows the base servicing various starships while the Enterprise departs from the facility.