According to the Wikipedia page: http://en.wikipedia.org/wiki/Fairfield_State_Hospital
The records from Fairfield Hills Hospital are located in RG 021, subgroup 7 at the Connecticut State Library & Archives in restricted files arranged into three series:
Patients, 1933-1995 admission and discharge records;
Personnel, 1934-1943 records of separation; and
Medical Staff Records, 1937-1974 records of birth and deportation, autopsies, and medical examiner cases.
I dont know what restrictions are on the files, you`ll have to check with the CT State Library.