Brett is the Rodney Dangerfield of the Nostromo crew—the man gets no respect, no respect at all.
As an engineer, he's a member of the Nostromo's "lower class" along with Parker, meaning he gets paid less than the rest of the crew. He is also Parker's lackey. Although the two men are friendly co-workers, Parker is the alpha to Brett's beta. We see Brett's pushover attitude (or easygoing, depending on your perspective) in his very first scene, when Brett reminds Parker about the bonus situation only to have Parker actually confront Dallas and the others. Even Ripley gets on his case for agreeing with whatever anyone else says:
RIPLEY: Whenever [Parker] says anything, you say, "Right," Brett. You know that?
BRETT: Right. (Alien)
Unfortunately for Brett, this passive nature doesn't let him avoid conflict the way he'd prefer, and he becomes a victim of the fully grown alien.
When the crew are tracking the alien—still thinking it is a wee baby beastie—they accidentally pick up Jonesy on the tracker and fail to catch him. Parker says they have to get the cat to prevent picking it up on the motion tracker again. Brett replies, "I'll—I'll," and Parker finishes his thought for him, ordering, "Go and get it, man" (Alien).
Too passive to make his own decisions, Parker makes Brett's for him and the result is that Brett goes off on his own into a dark room. Need we say more?