He began his career in the late 1970s. A well-travelled player, Beasant's former clubs include Wimbledon, Newcastle United, Chelsea, Southampton, Nottingham Forest, Portsmouth, Tottenham Hotspur, Brighton & Hove Albion and Wigan Athletic. He is best remembered for his part in Wimbledon's 1988 shock FA Cup victory, during which he was the first goalkeeper to save a penalty in an FA Cup final. He made two appearances for the England national football team.
On 10 May 2015, Beasant became the oldest individual to be selected in the squad for a match in Football League history while representing Stevenage as a substitute against Southend in the League Two Play-Off semi-final second leg, at the age of 56.
He entered the Football League in 1979 at the age of 20 when his home-town club, Wimbledon, newly promoted to the Third Division, signed him from non-league Edgware Town. He made his debut for them against Blackpool on 12 January 1980 and played once again that season, in which Wimbledon were relegated. He became a regular player the following campaign when they were promoted and stayed loyal to the club even when they were relegated again in 1982, being a key player in the side that won the Fourth Division title in 1983, won promotion from the Third Division a year later, and completed a four-season rise to the First Division in 1986 when they gained promotion from the Second Division in only their ninth season as a Football League team.
He became the first goalkeeper to save a penalty in an FA Cup final when he blocked John Aldridge's spot-kick for Liverpool in 1988, and in doing so helped Wimbledon secure a famous 1–0 win. "...and he saved it - and made history - the first time ever when a penalty kick is not been converted in the FA Cup Final here", yelled BBC's John Motson into the microphone. He was also the first goalkeeper to captain his team in an FA Cup final. His ability to kick the ball some considerable distance fitted in well with the style of play Wimbledon were known for in the 1980s, nor was he afraid to move out of the area and upfield before kicking the ball, or to take free kicks. As a result, opposing defences could never relax when he had the ball. Beasant won two caps for England, and went to the 1990 World Cup as a late replacement when third-choice keeper David Seaman suffered a hand injury. Wimbledon also did well in the league after their promotion in 1986, going top of the First Division in early September before finishing sixth at the end of the 1986–87 season. They finished seventh in the cup winning campaign a year later.
However, the 1988 FA Cup final was the last game that Beasant played for Wimbledon. A month later he was sold to Newcastle United for £750,000 – a national record for a goalkeeper at the time.
Beasant's spell on Tyneside was short and ill fated as the Magpies struggled and were finally relegated from the First Division in bottom place, but Beasant had left in January 1989 to join Chelsea.
In September 1992, two mistakes in a match against Norwich City led to Chelsea manager Ian Porterfield telling the media Beasant would never play for the club again, although he returned to the side when Porterfield left later that season.
During the 1993–94 season, Beasant sustained an injury when he dropped a 2 kg bottle of salad cream (also reported as mayonnaise) on his foot, severing the tendon to his big toe. Due to his indisposition, he missed eight weeks of the season.
Following the arrival of new manager, Glenn Hoddle, who opted for Dmitri Kharine as his first choice keeper with Kevin Hitchcock in reserve, Beasant was unable to get back into the Chelsea squad and looked for a new club.
Beasant signed for Southampton in November 1993 for a fee of £300,000 to replace the recently departed Tim Flowers. Beasant made his debut in a 1–0 defeat at Everton on 4 December; despite a run of four defeats, his confidence gradually returned and he soon became a favourite with The Dell crowd. With the departure of Ian Branfoot and his replacement as manager by Alan Ball, the "Saints" eventually climbed out of the relegation zone, finishing the 1993–94 season one point above relegated Sheffield United.
At the start of the 1994–95 season, he was replaced by Bruce Grobbelaar but was restored as first-choice keeper for the last month of the season. Following Alan Ball's move to Manchester City in the summer of 1995, new manager Dave Merrington preferred Beasant in goal. The team struggled throughout the season, and were never far from the relegation zone, but finished level on points but with a better goal difference than Ball's Manchester City who were relegated. Beasant himself finished the season by being voted the club's Player of the Season.
For the 1996–97 season, Graeme Souness was appointed manager; initially, Souness kept faith with Beasant but after a series of injuries (during which Saints took Chris Woods on loan), Souness signed Maik Taylor from Barnet in January. Beasant's final first-team game for Southampton was a 1–0 defeat against Liverpool on 29 December 1996. Following the arrival of Paul Jones in the summer of 1997, Beasant was now only third-choice 'keeper, and after a loan move to Nottingham Forest in August 1997, the transfer was made permanent in November. In his four years at The Dell, he made a total of 105 appearances in all competitions.
In November 1997, he signed for Nottingham Forest at 38 years old, after a short period on loan. He spent four years at the City Ground, during which time they were relegated from the Premier League one season after promotion. He went on to become Forest's oldest ever player at 42. He produced arguably much of his best form during his time at the City Ground. He was solid and reliable, and consistently produced good saves each season.
He then signed for Portsmouth in August 2001 after their regular goalkeeper Aaron Flahavan was killed in a car crash.
He played his last competitive game in the 2002–03 season for Brighton & Hove Albion in Division One at the age of 43, although he did spend the 2003–04 season registered as a player with Fulham in the FA Premier League. By then he was the oldest player registered with any professional club and the last in England with a 1950s birthdate.
He came out of retirement on 17 August 2013 to play for Southern League Division One Central club North Greenford United in a 2-0 defeat against Chalfont St Peter.
Having joined Stevenage as the club's goalkeeping coach in the middle of 2014, Beasant was named as a substitute for an away match at Carlisle United on 11 October 2014, aged 55.
The first of Beasant's two England caps came on 15 November 1989 against Italy. The following month, on 13 December, he made his second and final England appearance against Yugoslavia. He was a member of England's 1990 FIFA World Cup squad, called up after David Seaman had to withdraw through injury.
By the time of his retirement, Beasant had been appointed as a goalkeeping coach at Fulham in addition to serving as goalkeeping coach for Northern Ireland under former Wimbledon teammate Lawrie Sanchez. Beasant resigned from the Northern Ireland post in 2007 after Sanchez was appointed Fulham manager only for the pair to both be sacked by the club in December 2007.
Beasant was a senior coach at the Glenn Hoddle Academy. He joined in 2008 and worked not only with the Academy's goalkeepers, but also outfield players, especially defenders.
In August 2012, Beasant was appointed part-time goalkeeper coach at Bristol Rovers.
In July 2014, he became goalkeeping coach at Stevenage where he joined his son Sam Beasant.
On 25 June 2015, he was appointed as goalkeeping coach at Reading.Wimbledon
FA Cup winner: 1988
Football League Third Division runner-up: 1983–84
Football League Fourth Division champion: 1982–83
Football League Group Cup runner-up: 1982
Full Members Cup winner: 1990
Football League First Division champion: 1997–98