Striker Marco Streller struck in the ninth minute after poor goalkeeping from David De Gea and Alexander Frei scored late in the game as the Swiss champions took the second qualifying place in Group C.
United pulled one back a consolation through Phil Jones one minute from time.
United striker Wayne Rooney continued his unhappy run with several misses in a generally lacklustre performance by the Premier League side, who were left to contemplate the unglamorous Europa League.
Their closest effort came when Basel defender Markus Steinhoefer crashed an attempted clearance against the underside of his own bar.
It was only the third time in 17 Champions League partipations that United had failed to get past the group stage.