The Associated Press, in this one particular case at least, gets right to the point:
Sen. Robert Menendez employed as an unpaid intern in his Senate office an illegal immigrant who was a registered sex offender, now under arrest by immigration authorities, The Associated Press has learned. The Homeland Security Department instructed federal agents not to arrest him until after Election Day, a U.S. official involved in the case told the AP.
Of course, Homeland Security is denying that they delayed the arrest; and Senator Menendez is denying that he had any idea that he had as an intern someone who was either an illegal immigrant or a sex offender. Mind you, what was Homeland Security going to say? Sure, we held up on that arrest because the Senator was already dealing with a hooker controversy anyway, and it didn't seem sporting to pile on? - Which is something that happens more times in Washington DC than people really want to admit, honestly. As for the Senator Menendez not knowing about the illegal immigrant thing... he may be fibbing. As an unpaid intern the fellow in question is not required to fill out an Form I-9 (which would have caught his status) if he received no remuneration whatsoever. And certainly Senator Menendez is not the sort to use E-Verify on his interns anyway.
So... speaking theoretically, Senator Menendez probably could have and definitely should have checked; but speaking practically, well, there's a reason why US Senators generally consider themselves to be above the more onerous requirements of the law, and that's largely because they've designed the system so that attitude happens to be true. But also speaking practically, I suspect that Menendez's office will be surreptitiously checking the immigration status of its interns for a good, long while. That 'registered sex offender' thing is a little awkward...
PS: Yeah, I know: if it was a Republican Senator we'd be hearing calls for him to resign already. I didn't make the system; I just report on it.