'engage' is the correct form. When you are doing these exercises it's often helpful to look at the grammar of the sentence, especially any prepositions before or after the gap. You can 'undertake' something (eg undertake a task) but not 'undertake in' You can 'involve yourself in' something or you can 'get/be involved in' something, but you can't just 'involve in' something. And, as Johnny says, 'enlist' is wrong because it has the meaning of 'sign up for' eg in the army.

That's a difficult question to answer. The meanings are slightly different for each word, but each word could also work in that specific sentence, except "involve". The judge said that those prepared to involve themselves in crime must be ... The judge said that those prepared to be involved in crime must be ... Of the four choices, "engage in" is my preference. "Enlist in crime" sounds like joining a club for an activity, and "undertake in crime" just doesn't sound natural. "Undertake a life in crime" sounds better, though I'm fairly certain that "undertake in crime" is grammatically acceptable.