(written from a Production point of view)
Archer met Becky in a restaurant in Chinatown in San Francisco in 2153, shortly after the Xindi probe attacked Earth. Archer, who was late for their rendezvous, was hoping to spend some time with Becky. Becky told Archer that she hoped he did not have a girl in every spaceport and the two kissed. After presuming aloud that Archer was hoping for her to invite him back to her apartment, Becky opened a fortune cookie and said that he was in luck.
Becky was played by Serena Scott Thomas.
Becky's scene, number 42 in the installment, was cut from the final episode due to the fact that the completed episode was ten minutes too long. The scene (along with the previous one) was included in the ENT Season 2 DVD.
In the final draft script of "The Expanse", this character's name was given as "Rebecca", with "Becky" being established as a shortened form. The wbm at StarTrek.com also referred to her as "Rebecca", as does the novelization of The Expanse. However, "Becky" is the only way she is referred to in dialogue (in both the script and the filmed version of the scene). The script also described her as "attractive".
Prior to the scene being cut, Archer actor Scott Bakula felt the Becky character had potential and that she was a way of establishing that Archer had a life outside of his career. Bakula commented, "I think there is a possibility we may see her again, depending on when we get back to Earth." He thought it unlikely that Becky would end up marrying Archer, when the captain returned, but suspected their relationship "could go a lot of ways" and "open up a million things" regarding the narratives of Star Trek: Enterprise. (Star Trek: Communicator issue 145, pp. 42 & 45)
The novelization of The Expanse suggests that, since her last encounter with Archer, Becky had changed her hairstyle, with it having previously been a different length than shoulder-length and less red. Additionally, the novelization states that Becky's mother was a significant figure in the early history of Porthos.