Started in the 1960s.
The plan was set in motion by the Community Mental Health Act, passed by Congress in 1963, mandating the appointment of a Commission to make recommendations for “combating mental illness in the United States”
http://en.wikipedia.org/wiki/Deinstitutionalisation
(I know it is wikipedia, but the article seems accurate, from what I know)
In my state, I recall it being a topic of conversation in the mid-1970s.
To be fair, I remember some of the hospitals for the mentally ill in the 1950s and they were awful. I volunteered in one the summer of 1958 and it was just depressing. The people who worked there, for the most part, were pretty desensitized, themselves. OTOH, the community-based options haven’t worked, either.
When I was in the Army, I spent 6 weeks as an observer in a mental hospital in 1973, in Washington State, it wasn’t too bad but the staff did seem a little neurotic.