Are snatch pickups used today? John T. Downey, who spent 20 years imprisoned in China, was captured when his plane was shot down during an attempted "snatch pickup". The spy they were supposed to retrieve had been turned by the Chinese, and they were waiting for the plane with anti-aircraft weaponry. This 1951 attempt was the first time the snatch pickup was tried in a war, according to this article about John T. Downey:


  The snatch pickup was a bizarre maneuver, and an untested one. The
  process called for an aircraft flying at low altitude to hook a line
  elevated between two poles. Connected to the line would be a harness,
  into which an agent would be strapped. The contraption resembled a
  swingset, if that swingset was designed to be born aloft by a cargo
  plane. Two men in the back of the plane would operate a pulley,
  dropping the hook intended to catch the agent and then reeling in the
  line; both the operators and the pilots required extensive training.
  Describing the operation in his 1984 book Perilous Missions, a history
  of CIA covert operations in Asia, William M. Leary writes that the
  agent was forced to sit impassively in his harness, awaiting possible
  decapitation, among a litany of other potential injuries. The pickup
  of Downey’s courier would be the first time the CIA ever carried the
  plan into action. The Air Force deemed the operation too risky to try.


Whatever happened to this technique? Is it used today?
No, although they were used up into the 1990s. Wikipedia has an article on this - called the Fulton Recovery system. Your quoted paragraph is incorrect - U.S. Air Force Special Operations Command (AFSOC) did use the system, on modified MC-130s. Despite the danger, Wikipedia only mentions one death using the system, though given the clandestine nature of many of the missions it was used on, it would not be surprising if there were other as yet unrecorded instances of its use.

Modern special operations helicopters have a much greater range, and some, like the MH-53, MH-47, and MH-60, are actually air-refuelable, extending their range. Additionally, the CV-22 Osprey has a much greater range and speed than a traditional helicopter. These improvements in extraction vehicles made the 'skyhook' system obsolete. 

Note that this article mentions the British used the system in 2001 to rescue an injured soldier behind enemy lines.