The 33-year-old capitalised on slack marking from a corner to bundle in the winner on 52 minutes and dent the Reds' hopes of reaching the Champions League knockout stage
Streller capitalised on some slack marking from a corner in the 52nd minute to bundle in the only goal of a game which saw the Reds uninspired in the first half and profligate in the second.
Raheem Sterling and Lazar Markovic both spurned presentable opportunities as the Reds pushed forward in search of an equaliser but Brendan Rodgers's men now go into two matches with Real Madrid level on points with the Swiss champions.
Liverpool made two changes to their starting line-up from Saturday's Premier League derby draw against Everton, with Jose Enrique replacing Alberto Moreno and Philippe Coutinho coming back in for Adam Lallana.
Basel went into the match with problems at the back, with Marcelo Diaz suspended and fellow defenders Philipp Degen, Ivan Ivanov and Walter Samuel all sidelined through injury. Paulo Sousa also preferred Breel Embolo to Derlis Gonzalez up front, even though the Paraguay international was on target in defeat at the Santiago Bernabeu.
Sterling had the ball in the net in the opening stages but the flag was correctly up for offside, while Basel had to make an early change when Behrang Safari limped off to be replaced by Gonzalez.
Liverpool survived a scare when Gerrard was dispossessed in front of his own box, while Gonzalez was twice guilty of a poor touch when getting the wrong side of Enrique.
Basel continued to show plenty of pace on the break, with Simon Mignolet saving well with his legs from a Serey Die shot, and Enrique caused problems when getting forward for Liverpool but it remained goalless at the break.
The home fans were appealing for a penalty when Sterling appeared to bundle over Gonzalez in the area at the start of the second half and then Mignolet had to get down well to save from Ahmed Hamoudi on the break.
However, Mignolet could only palm out a deflected header from the resulting corner and veteran striker Streller pounced on the loose ball to score from close range after 52 minutes.
Basel goalkeeper Tomas Vaclik only just dealt with a Mario Balotelli free kick from fully 30 yards out and Sterling was later put clean through following a neat build-up, but could not control the pass.
Rickie Lambert replaced Markovic as Liverpool went in search of an equaliser but Gerrard's late free kick failed to really test Vaclik and Balotelli was offside when netting as the hosts held on.
Basel, who had to play their last European home match behind closed doors because of crowd trouble, could face more disciplinary action, though, after referee Jonas Eriksson reported an object thrown onto the pitch.