To see New York State, check Pt. 34 in the 1960 Census of Population Vol. 1; Characteristics of the Population; Number of Inhabitants; General Population Characteristics; General Social & Economic Characteristics; Detailed Characters.
To see 1960 Census of Population, General Social & Econimic Characteristics - Ethnic origin, migration famiies, income, education, employment, etc (33C-36C), check out Volumes 33C-36C. For New York State, check out Volume 34C.
The National Historical Geographic Information System (NHGIS), produced by the Minnesota Population Center at the University of Minnesota provides free access to historical Census information. Note: this resource requires user registration, but registration is free and immediate.