In 2353, when Jack Crusher died aboard the USS Stargazer, Captain Picard came to this base to meet with Jack's widow, Beverly Crusher, and present the body to her for funeral. His body was stored in the base's morgue. (TNG: "Violations")
This starbase's quadrant of origin is inferred based on its location in the Maxia sector. The namesake of this sector, the Maxia system, was located in the Alpha Quadrant as seen in the star chart appearing in the Star Trek: Picard episode "Maps and Legends".
Starbase 32 was not explicitly named in the episode or script for "Violations" but comes from a placard seen on the wall behind Picard during the flashback sequence. This was created by the show's art department and features the names of various Star Trek production staff. The bottom of the plaque reads "Starfleet Operations Support Services" suggesting Starbase 32 was assigned to this department. (Star Trek: The Next Generation - The Continuing Mission, p.157)
The set for the sections of Starbase 32 seen in this episode were built on Paramount Stage 16.
Starbase 32 appears in a number of non-canon works.
The Star Trek: The Lost Era novel Well of Souls tells the story of Lieutenant Commander Darya Bat-Levi (β) and her family who live on the station during the early 24th century. During a test run of a modified shuttle, Bat-Levi is seriously injured and is admitted to the station's medical facilities for treatment. The station is also mentioned as having a team of psychiatrists aboard.
According to the Pocket TNG novel The Buried Age, Admiral Naomi Jeruslami (β) takes up command of the station in 2354. The following year, Captain Picard's court of inquiry (β) over the death of Jack Crusher on the Stargazer takes place on the station.
Starbase 32 is also mentioned in Metamorphosis.