In 1952 Stevenson was appointed Recorder of Cambridge (a part time judge). In 1954 he represented the British Government in Kenya during Jomo Kenyatta's unsuccessful appeal against his conviction for involvement in Mau Mau activities.
In 1955, aided by junior counsel Sebag Shaw and Peter Rawlinson, he defended Ruth Ellis. His conduct of the defence has been severely criticised by writer Monica Weller for giving the "prosecution an easy time, subjecting prosecution witnesses to a minimum of cross-examination". He opened by saying: "Let me make this abundantly plain: there is no question here but this woman shot this man [...] You will not hear one word from me – or from the lady herself – questioning that." The jury took 23 minutes to find Ellis guilty. She was hanged.
Stevenson was Sir Reginald Manningham-Buller's number 2 during the failed prosecution of suspected serial killer John Bodkin Adams in 1957. The prosecution's conduct has been heavily criticised, and its decision to drop a second murder charge via a nolle prosequi was later deemed by the presiding judge, Patrick Devlin, "an abuse of process". It has been said that the prosecution might have had a better chance of success if Stevenson had been allowed to lead. He was appointed a High Court judge on 1 October 1957, and (as is traditional) knighted a few days later. For the first four years he was assigned to the Probate, Divorce and Admiralty Division, and it was only after his transfer to the Queen's Bench Division, that he began to attract press attention.
In 1969, as a judge, he sentenced the Kray brothers, saying "In my view, society has earned a rest from your activities." Then in 1970 he controversially gave long sentences to Cambridge University students for taking part in demonstrations against the military government in Greece.
He died in 1987 of a heart attack.
When presiding over a rape case, he described it as a "pretty anaemic kind of rape" because the accused's ex-girlfriend was the victim.
Stevenson once said of a husband in a divorce case, “He chose to live in Manchester, a wholly incomprehensible choice for any free man to make“.
To a man acquitted of rape he commented, “I see you come from Slough. It is a terrible place. You can go back there“.
Of the Krays, he once said that they had only told the truth twice during the trial - when Reggie referred to a barrister as "a fat slob" and later when Ronnie accused the judge of being biased.
In 1978 Stevenson was reprimanded by the Lord Chancellor Elwyn Jones for calling the Sexual Offences Act 1967 a "buggers' charter".
In the 1994 memoirs, Sword and Wig, of retired Court of Appeal judge, Sir Robin Dunn, Stevenson was described as, "the worst judge since the war". This prompted a number of high profile legal figures to defend Stevenson.