Born in Lambeth, London, Babb came through the youth ranks of Millwall, before beginning his senior career in 1990 with Bradford City. He moved to Coventry City in July 1992 for a fee of £500,000, where he spent two seasons before moving to Liverpool on 1 September 1994 for £3.6million, which made him the most expensive defender in Britain at the time.
Having chosen to represent the Republic of Ireland internationally, Babb earned 35 full caps, including four matches at the 1994 FIFA World Cup. In August 2000, he and Ireland teammate Mark Kennedy were sent home from a training camp "after appearing in court charged with drunken and abusive behaviour and causing criminal damage."
Babb's final game for Ireland was in UEFA Euro 2004 qualifying against Russia in Moscow, on 7 September 2002: having come into the game in the 85th minute, his first touch of the ball was diverted behind Irish goalkeeper Shay Given for an own goal.