Crothers grew up in Cookstown and studied at Rainey Endowed School, where he won the top mathematics and Science State Exhibition prize in 1959. He then read mathematics at Balliol College, Oxford, before obtaining a doctorate from the Queen's University of Belfast in 1966. In the following years, he lectured at Queen's, undertook research at University College London, worked as a tutor for the Open University, and was elected as a Fellow of the Institute of Physics.
At the 1973 Northern Ireland Assembly election, Crothers was elected for the Alliance Party of Northern Ireland in South Antrim, receiving 5,975 first preference votes. He did not stand for the Northern Ireland Constitutional Convention or subsequent assemblies, but did contest the Lower Falls area of Belfast City Council at the 1989 local elections. He took only 135 votes and was not elected.
In 1985, Crothers was promoted to a Personal Chair in Theoretical Physics at Queen's. He was elected Member of the Royal Irish Academy in 1991 and Fellow of the American Physical Society in 1994, before retiring in 2007.