He was born in Royal Leamington Spa, Warwickshire, and educated at Leamington College and Downing College, Cambridge. After graduating he held a number of academic posts outside the United Kingdom: in Egypt, Japan, Thailand and notably in Singapore (from 1960). He at times attributed his lack of success in finding a post closer to home to writing for Scrutiny and his short association with F. R. Leavis; whose influence he mainly and early, but not entirely, rejected.
As a poet he was identified with the Movement. His 1955 anthology, Poets of the 1950s, served to delineate the group of British poets in question — albeit somewhat remotely and retrospectively, since he was abroad and it was not as prominent as the Robert Conquest collection New Lines of the following year.
Returning to London in 1970, he edited Encounter magazine, with Melvin J. Lasky, for two years. He subsequently worked in publishing.
Enright gained some notoriety in Singapore after his inaugural lecture at the University of Singapore on 17 November 1960, titled "Robert Graves and the Decline of Modernism". His introductory remarks on the state of culture in Singapore were the subject of a Straits Times article "'Hands Off' Challenge to 'Culture Vultures'" the next day. Among other things, he had said that it wasimportant for Singapore and Malaya to remain "culturally open", that culture was something to be left for the people to build up, and that for the government to institute "a sarong culture, complete with pantun competitions and so forth" was futile. Some quotes include:
Art does not begin in a test-tube, it does not take its origin in good sentiments and clean-shaven, upstanding young thoughts.
Leave the people free to make their own mistakes, to suffer and to discover. Authority must leave us to fight even that deadly battle over whether or not to enter a place of entertainment wherein lurks a juke-box, and whether or not to slip a coin into the machine.
The following day, Enright was summoned by the Ministry for Labour and Law regarding his foreigner work permit, and was handed a letter there by the Minister for Culture S. Rajaratnam, which had also been released to the press. This letter admonished Enright for "involv[ing] [himself] in political affairs which are the concern of local people", not "visitors, including mendicant professors", and said that the government "ha[s] no time for asinine sneers by passing aliens about the futility of 'sarong culture complete with pantun competitions' particularly when it comes from beatnik professors." There was also some criticism that Enright had been insensitive towards Malays and their so-called "sarong culture."
With some mediation from the Academic Staff Association of the university, it was agreed that to put the matter to rest, Enright would write a letter of apology and clarification, the government would reply, and both were to be printed in the newspapers. Although the affair was "essentially dead" after that, according to Enright, it would still be brought up periodicially in discussions of local culture and academic freedom.
Enright gives his account of the incident in his Memoirs of a Mendicant Professor (pp. 124-151).