Josephson was born in Cardiff, Wales, to Jewish parents, Mimi (née Weisbard, 1911–1998) and Abraham Josephson. He attended Cardiff High School, where he credits some of the school masters for having helped him, particularly the physics master, Emrys Jones, who introduced him to theoretical physics. In 1957 he went up to Cambridge, where he read mathematics at Trinity College, Cambridge. After completing Maths Part II in two years, and finding it somewhat sterile, he decided to switch to physics.

Josephson said that Trinity College's long interest in the paranormal meant that he did not dismiss these ideas out of hand. Several Presidents of the Society for Psychical Research had been fellows of Trinity, and the Perrott-Warrick Fund, set up in Trinity in 1937 to fund parapsychology research, is still administered by the college. He continued to explore the idea that there is intelligence in nature, particularly after reading Fritjof Capra's The Tao of Physics (1975), and in 1979 took up a more advanced form of TM, known as the TM-Sidhi program. According to Anderson, the TM movement produced a poster showing Josephson levitating several inches above the floor. Josephson argued that meditation could lead to mystical and scientific insights, and that, as a result of it, he had come to believe in a creator.

He graduated in 1960 and became a research student in the university's Mond Laboratory on the old Cavendish site, where he was supervised by Brian Pippard. American Physicist Philip Anderson, also a Future Nobel Prize laureate, spent a year in Cambridge in 1961–1962, and recalled that having Josephson in a class was "a disconcerting experience for a lecturer, I can assure you, because everything had to be right or he would come up and explain it to me after class." It was during this period, as a PhD student in 1962, that he carried out the research that led to his discovery of the Josephson effect; Cambridge unveiled a plaque on the Mond Building dedicated to the discovery in November 2012. He was elected a fellow of Trinity College in 1962, and obtained his PhD in 1964 for a thesis entitled Non-linear conduction in superconductors.

Before Anderson and Rowell confirmed the calculations, the American Physicist John Bardeen, who had shared the 1956 Nobel Prize in Physics (and who shared it again in 1972), objected to Josephson's work. He submitted an article to Physical Review Letters on 25 July 1962, arguing that "there can be no such superfluid flow." The disagreement led to a famous confrontation in September that year at Queen Mary College, London, at the Eighth International Conference on Low Temperature Physics. When Bardeen (then one of the most eminent physicists in the world) began speaking, Josephson (still a student) stood up and interrupted him. The men exchanged views, reportedly in a civil and soft-spoken manner.

Josephson spent a postdoctoral year in the United States (1965–1966) as research assistant professor at the University of Illinois at Urbana–Champaign. After returning to Cambridge, he was made assistant Director of research at the Cavendish Laboratory in 1967, where he remained a member of the Theory of Condensed Matter group, a theoretical physics group, for the rest of his career. He was elected a Fellow of the Royal Society (FRS) in 1970, and the same year was awarded a National Science Foundation fellowship by Cornell University, where he spent one year. In 1972 he became a reader in physics at Cambridge and in 1974 a full professor, a position he held until he retired in 2007.

Josephson was awarded several important prizes for his discovery, including the 1969 Research Corporation Award for outstanding contributions to science, and the Hughes Medal and Holweck Prize in 1972. In 1973 he won the Nobel Prize in Physics, sharing the $122,000 award with two other Scientists who had also worked on quantum tunnelling. Josephson was awarded half the prize "for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effects."

In the early 1970s Josephson took up transcendental meditation and turned his attention to issues outside the boundaries of mainstream science. He set up the Mind–Matter Unification Project at the Cavendish to explore the idea of intelligence in nature, the relationship between quantum mechanics and consciousness, and the synthesis of science and Eastern mysticism, broadly known as quantum mysticism. Those interests have led him to express support for topics such as parapsychology, water memory and cold fusion, and have made him a focus of criticism from fellow Scientists.

Winning the Nobel Prize in 1973 gave him the freedom to work in less orthodox areas, and he became increasingly involved – including during science conferences, to the irritation of fellow Scientists – in talking about meditation, telepathy and higher states of consciousness. In 1974 he angered Scientists during a colloquium of molecular and cellular biologists in Versailles by inviting them to read the Bhagavad Gita (5th – 2nd century BCE) and the work of Maharishi Mahesh Yogi, the founder of the TM movement, and by arguing about special states of consciousness achieved through meditation. "Nothing forces us," one scientist shouted at him, "to Listen to your wild speculations." Biophysicist Henri Atlan wrote that the session ended in uproar.

A practitioner of transcendental meditation (TM) since the early seventies, Josephson became a visiting faculty member in 1975 of the Maharishi European Research University in the Netherlands, part of the TM movement. He also held visiting professorships at Wayne State University in 1983, the Indian Institute of Science, Bangalore in 1984, and the University of Missouri-Rolla in 1987.

In 1976 Josephson travelled to California to meet two leading members of the group, laser physicists Russell Targ and Harold Puthoff, authors of Mind Reach (1977). Targ and Puthoff had set up a parapsychology ("psi") lab at SRI and had had papers published about their work – which included testing later-discredited claims by Uri Geller that he could make objects move using psychokinesis – in Nature and other peer-reviewed journals. The San Francisco Chronicle covered Josephson's visit.

Josephson co-organized a symposium on consciousness at Cambridge in 1978, publishing the proceedings as Consciousness and the Physical World (1980), with neuroscientist V. S. Ramachandran. A conference on "Science and Consciousness" followed a year later in Cordoba, Spain, attended by physicists and Jungian psychoanalysts, and addressed by Josephson, Fritjof Capra and David Bohm (1917–1992).

Science Writer Martin Gardner criticized Josephson in 1980 for complaining to the New York Review of Books, along with three other physicists, about an article by J. A. Wheeler that ridiculed parapsychology. Several physicists complained in 2001 when, in a Royal Mail booklet celebrating the Nobel Prize's centenary, Josephson wrote that Britain was at the forefront of research into telepathy. Physicist David Deutsch said the Royal Mail had "let itself be hoodwinked" into supporting nonsense, although another Physicist, Robert Matthews, suggested that Deutsch was skating on thin ice given the latter's own work on parallel universes and time travel.

By 1996 he had set up the Mind–Matter Unification Project at the Cavendish Laboratory to explore intelligent processes in nature. In 2002 he told Physics World: "Future science will consider quantum mechanics as the phenomenology of particular kinds of organised complex system. Quantum entanglement would be one manifestation of such organisation, paranormal phenomena another."

In 2004 Josephson criticized an experiment by the Committee for Skeptical Inquiry to test claims by Russian schoolgirl Natasha Demkina that she could see inside people's bodies using a special kind of vision. The experiment involved her being asked to match six people to their confirmed medical conditions (plus one with none); to pass the test she had to make five correct matches, but made only four. Josephson argued that this was statistically significant, and that the experiment had set her up to fail. One of the researchers, Richard Wiseman, professor of psychology at the University of Hertfordshire, responded that Josephson had no record of publishing on parapsychology. Keith Rennolis, professor of applied statistics at the University of Greenwich, supported Josephson's position, asserting that the experiment was "woefully inadequate" to determine any effect.

Josephson delivered the Pollock Memorial Lecture in 2006, the Hermann Staudinger Lecture in 2009 and the Sir Nevill Mott Lecture in 2010.

Matthew Reisz wrote in Times Higher Education in 2010 that Josephson has long been one of physics' "more colourful figures." His support for unorthodox causes has attracted criticism from fellow Scientists since the 1970s, including from Philip Anderson. Josephson regards the criticism as prejudice, and believes that it has served to deprive him of an academic support network.

Josephson's reputation for promoting unorthodox causes was cemented by his support for the ideas of water memory and cold fusion, both of which are rejected by mainstream Scientists. Water memory is purported to provide an explanation for homeopathy; it is mostly dismissed by Scientists as pseudoscience, although Josephson has expressed support for it since attending a conference at which French immunologist Jacques Benveniste first proposed it. Cold fusion is the hypothesis that nuclear reactions can occur at room temperature. When Martin Fleischmann, the British Chemist who pioneered research into it, died in 2012, Josephson wrote a supportive obituary in the Guardian and complained to Nature that its obituary had failed to give Fleischmann due credit. Antony Valentini of Imperial College London withdrew Josephson's invitation to a 2010 conference on the de Broglie-Bohm theory because of his work on the paranormal, although it was reinstated after complaints.

He has repeatedly criticized "science by consensus," arguing that the scientific community is too quick to reject certain kinds of ideas. "Anything goes among the physics community – cosmic wormholes, time travel," he argues, "just so long as it keeps its distance from anything mystical or New Age-ish." Referring to this position as "pathological disbelief," he holds it responsible for the rejection by academic journals of papers on the paranormal. He has compared parapsychology to the theory of continental drift, proposed in 1912 by Alfred Wegener (1880–1930) to explain observations that were otherwise inexplicable, which was resisted and ridiculed until evidence led to its acceptance after Wegener's death.