The ASR is most certainly still awarded.
The ol ASR (and a parachutist badge) was all I had for almost 2 years. Received an AAM during an exercise in Thailand, Good Conduct Medal a year after that, and an ARCOM when I left active duty after 4 years. 3 medals and 1 ribbon for 4 years of peacetime service.
Of course, 9/11 happened 2 weeks after my ETS, and the awards and op tempo (and the funerals) all increased dramatically.
II stand corrected. When the ASR first came out - it was when the NDSR was no longer authorized (ie no combat). My thinking was in essence was that the ASR was the peacetime equivalent of the NDSR. But as I had stated before - it is awarded upon completion of MOS - Combat or lack thereof is not a factor..