Well, the Captain is responsible for everything that is going on and he/she has to ensure that the preflight-check on an airliner is done, but he/she can delegate that duty to the SFO or FO.
Regarding "Trust, but verify": We drain our fuel tanks every time before we go on a flight. Leaving our home base in the early morning this seems like a pretty logical thing to do, especially when you keep in mind that the fueler sometimes forgets to refuel your aircraft (we have about ten aircraft and more coming home in the evening at the same time, so it happens that the fueler oversees one). Anyway, when we stop on a cross-country, have a break and get some fuel, we drain again. I never ever noticed something unusual and you would think everything is okay, cause you already drained in the morning and I have often asked me why we have to do that every time again and again. How suprised was I, when I found out on our break that we had quite some water in the one tank today! This really shows that you can not trust something because it seemed to have worked so far, but need to check again every single time...
Cheers,
Felix