As with the rest of the Star Trek films, The Final Frontier was produced by Paramount Pictures—the same company whose TV division launched the original series. Star Trek and Paramount go way back.
There are few production companies with as illustrious a history as Paramount Pictures. The organization was founded in 1912, making it the fifth-oldest film studio in the world, and it consistently released groundbreaking films throughout the 20th century.
As mentioned, Paramount has been running the Star Trek show since the beginning. Although the original series was a bomb, lasting only a few seasons, the company found the film series to be a whopping financial success. In fact, Star Trek IV had just proven to be the most profitable of the bunch.
So, of course, they were going to make another film. Although there was some slight tension between producers and first-time director (and star) William Shatner, The Final Frontier was free of the tense work environment that had defined earlier Star Trek films. By this point, the series was running like a well-oiled dilithium crystal.