After a deflected Nacho effort put Real 1-0 ahead in the 14th minute, Bale raced on to a Luka Modric pass and dinked the ball brilliantly over onrushing Basel goalkeeper Tomas Vaclik before volleying into an empty net.
A minute later, Bale found space on the right of the penalty area and sent over a low cross for Cristiano Ronaldo, last season's Champions League top scorer with 17 goals, to tap in at the far post.
New Real signing James Rodriguez made it 4-0 in the 37th minute when he started and finished a sweeping counter attack before Derlis Gonzalez almost immediately pulled a goal back for the visitors when he sped clear and beat Iker Casillas with a low strike.
Karim Benzema completed the scoring in the 79th minute when he lashed a left-foot shot into the net off the underside of the crossbar as Real became the first club to score 1,000 goals in European competition.
Real, chasing a record-extending 11th European crown, came into the game at the Bernabeu having lost twice on the trot in La Liga and were under pressure to produce a convincing performance in front of their demanding fans.
While Basel hardly represent the stiffest test they will face as they seek to become the first team to defend their European title in the Champions League era, the victory will lift spirits ahead of Sunday's La Liga game at promoted Deportivo La Coruna.
Real coach Carlo Ancelotti has had problems integrating Rodriguez and fellow new signing Toni Kroos in midfield but both produced solid displays against the Swiss champions, with Rodriguez supplying a clever back heel to send Nacho clear in the buildup to the opening goal.
Despite the drubbing, Basel played some attractive football of their own and almost scored a second in the 76th minute when Fabian Schaer struck a curling effort against a post after a neat passing move through the middle.