I don't immediately recall the precise origin
Like some of his other important ideas, Bernstein's presentation has mostly been disseminated through the papers of other people. Probably the affine Iwahori-Hecke algebramost influential is the 1989 JAMS paper by Lusztig, but two freely available from the AMS here. Combinatorial work by Arun Ram and others involving affine Hecke algebras also depends on this viewpoint: see for example Parkinson-Ram
Two relevant recent papers with extensive references are also available on arXiv and would be worth looking at in any case: haines-pettit and goertz.
