Born in Buffalo, New York, Crotty received a B.A. from the University of Notre Dame in 1962 and an LL.B. from Cornell Law School in 1967. He was also in the United States Navy Reserve from 1962 to 1968. He was a law clerk to the Hon. Lloyd F. MacMahon, U.S. District Court for the Southern District of New York, from 1967 to 1969. Crotty entered private practice in New York City at the prominent law firm of Donovan, Leisure, Newton & Irvine, working there as an associate from 1969 to 1976, and then as a partner from 1976 to 1984, and from 1988 to 1993. From 1984 to 1988, he held several government positions in the Office of Financial Services for the City of New York. He was a commissioner in that office in 1984, and then Commissioner of Finance from 1984 to 1986, and a commissioner for Housing Preservation & Development until 1988. After having returned to private practice for a time, he became the Corporation Counsel at the New York City Law Department from 1994 to 1997, and group president for the New York and Connecticut region for Verizon Communications from 1997 to 2005.