How much responsibility does ATC have to alert a flight about being in a dangerous situation? There have been quite a number of accident reports referenced here at Aviation.SE about commercial (passenger or freight) flights executing controlled flight into terrain on approach for landing. The accident reports always seem to revolve around the pilots not knowing they are too low, failing to recognize or respond to the GPWS, etc.

While I understand that the pilot in command has the final say on what happens on board the aircraft, ATC is responsible for giving the PIC instructions to get him safely to the end of the runway. 

In the case where ATC has given the PIC the approach route, whether VFR (flight following) or IFR, how much time, ability, and requirement does ATC then have to monitor that flight to ensure that it's on the expected course and at expected altitudes? In other words, how much responsibility does ATC have for failure to monitor the situation and allow these accidents to happen?

If necessary, differentiate between different CAAs, but I would think the answer would be fairly similar around the world.

There are a couple of nice answers, but they're not addressing (in my head, at least) the question I was attempting to ask, so it seems I didn't ask it well. I'll clarify further:

In a situation such as Eastern 401, the pilots had already contacted the tower to indicate that they had an issue - the nose gear lock light failed to illuminate. While attempting to determine what the cause of the problem was (bulb burnt out or gear not actually locked), they forgot to aviate and flew into the ground. In a situation like this, particularly where ATC has been notified that there was an issue with the flight, it seems to me that they (ATC) would have been more alert to the flight and notified the pilots that they were descending when they should have been holding a steady altitude. 

This is the type of situation that I'm asking about (though not specifically EA401), and I'm wondering what role, if any, ATC should or does take in monitoring and alerting flight crews to altitude deviations, especially on approach.
This is potentially a very broad topic, but here's some information about the US.

As a general principle, safety is ATC's first priority so obviously they have a basic responsibility to provide whatever information and action are required to keep aircraft safe:

FAA ATC orders 2-1-1:


  The primary purpose of the ATC system is to prevent a collision
  between aircraft operating in the system and to provide a safe,
  orderly and expeditious flow of traffic, and to provide support for
  National Security and Homeland Defense.


The orders explicitly say that ensuring separation and issuing alerts take priority over everything else. Section 2-1-2, DUTY PRIORITY:


  Give first priority to separating aircraft and issuing safety alerts
  as required in this order. Good judgment must be used in prioritizing
  all other provisions of this order based on the requirements of the
  situation at hand.


Section 2-1-6, SAFETY ALERT defines what an alert is and when to issue one (this section is referenced throughout the orders):


  Issue a safety alert to an aircraft if you are aware the aircraft is
  in a position/altitude that, in your judgment, places it in unsafe
  proximity to terrain, obstructions, or other aircraft. Once the pilot
  informs you action is being taken to resolve the situation, you may
  discontinue the issuance of further alerts. Do not assume that because
  someone else has responsibility for the aircraft that the unsafe
  situation has been observed and the safety alert issued; inform the
  appropriate controller


It's difficult to say much more than that because there are so many possible scenarios. But you can search for "safety alert" in the orders and the AIM to get more information and there's some limited content in the orders 5-15 about radar displays and alerts.

If ATC fails to issue a safety alert and that leads to an accident, then it will be mentioned in the NTSB report, e.g. LAX04FA205:


  Contributing to the accident was the failure of both the Center and
  TRACON controllers to properly respond to the aural and visual Minimum
  Safe Altitude Warning System (MSAW) alert.


But actually acting on a safety alert is entirely the pilot's responsibility, of course (this is from the AIM 4-1-16):


  Once the alert is issued, it is solely the pilot’s prerogative to
  determine what course of action, if any, to take