Terminal controllers with ARTS or STARS systems have MSAW capability. Be aware though, that some small VFR (no approach control) facilities have tower displays which are for "informational" use only. Those controllers are allowed to use the tower display as an aid to the eyes, and for issuing safety alerts, but not allowed to vector etc. These facilities may not have MSAW capability or aural alarms.
ATC is indeed responsible for separation from terrain. However, it is a shared responsibility. Always be aware of your altitude reference terrain and obstructions. I used to work in a mountainous area, and daily descended aircraft for ILS turn-ons at an altitude which was below terrain just a dozen miles beyond the localizer. Had an aircraft gone NORDO at just the wrong time, could have been exciting. In fact, it DID happen several times, but always when WX was good.
I also vividly recall many pilots reporting the field in sight from many miles out at night, issuing a visual approach clearance, and then having to issue an urgent terrain warning to aircraft descending below safe altitude 20-25 miles from the airport. Likewise, many pilots scared themselves silly when they got a GPWS alert from unseen ridges and rocks a few miles from the airport. These were not just amatuers, but some Air Carrier Pilots as well. Apparently some folks just descend right down to the FAF/GS intercept altitude 20 miles out if they've been cleared for a visual. Not a good idea in many places.
In fact, if any of you are familiar with the Loc Rwy 4 approach to ELP, you know it's a fairly steep approach beginning over the Juarez mountains, and continuing closely parallel to the south edge of the Franklin Mountains just west of the airport. In very bad WX, it was NOT unusual for every third or fourth aircraft so equipped to get a GPWS alarm and abandon the approach as they crossed the Rio Grande and descended near the ridge. It was also not unusual for our own MSAW to go off on nearly EVERY approach. We seldom broadcast the advisory, but checked the course and altitude of the inbound to make sure it appeared normal. Still, and alarm that goes off on nearly every approach is a nusance alarm, and detrimental to safety. The alarm was usually set off by the aircraft's rate of descent, (dive and drive?) and not the actual altitude for the approach segment.
Back to the matter that began the thread, it's impossible to say without knowing more, but quite possible the controller issued a descent to MVA for the area the aircraft was IN, (which in mountainous areas can be several thousand feet below MEA, depending on radar coverage), and then realised that altitude was not appropriate to the route further along. Or perhaps just screwed up. But that's what pilots and controllers are SUPPOSED to do, watch each other's backs. FSDO and NTSB get to assign blame and responsibility after the fact. We're supposed to be PREVENTING/CORRECTING errors before they get noticed by FSDO or NTSB.