Gene Roddenberry questioned Carey Wilber's notion of wasting a high-tech spaceship and expensive resources on criminals - just like Kirk and Spock came up with the same question in the story itself - and came up with the concept of "a bunch of Napoleons" self-exiling in space.
Being a first-season episode, Chekov (Walter Koenig) does not appear. Nevertheless, Chekov does appear in the film Star Trek II: The Wrath of Khan (1982), in which Khan not only meets but instantly recognizes him. Many fan theories subsequently tried to explain where Chekov could have been off-screen during that episode that would cause Khan to remember him. Walter Koenig himself came up with a story, which he likes to recite at conventions, that Khan, during the events of Space Seed, desperately needed to go to the bathroom, but the only toilet he could find was occupied, and when it was opened, Chekov walked out and Khan resolved never to forget Chekov's face. The Wrath of Khan novelization by Vonda N. McIntyre does officially explain that Chekov was working in Engineering when Khan began his rebellion there (and most of that happened off-camera), and it was because of Chekov's valiance in resisting that he was promoted to the Bridge for the series' second season. See also trivia for Star Trek Into Darkness (2013).
The automated signal that the S.S. Botany Bay sends, "CQ", is an actual Morse code shorthand for "seek you", or "please answer, anyone that can hear this."
The main cast were enthusiastic about working with Ricardo Montalban. DeForest Kelley later said "I enjoyed working with Ricardo the best. I was privileged. He is a marvellous actor."
Carey Wilber used the 18th century British custom of 'transportation' (shipping out convicts to the colonies, especially Australia) as a parallel for his concept of "seed ships", used to take unwanted criminals out to space from the overpopulated Earth (hence the name Botany Bay). In his original treatment, the Botany Bay left Earth in 2096, with 100 criminals (both men and women) and a team of several volunteering lawmen aboard.