Draxler's beauty lifts Schalke
Draxler struck a rising shot from 20 yards in the 54th minute, soon after Roman Neustaedter's shot from the same range cannoned off the Basel crossbar.
Schalke's second victory in Group E group halted Basel's momentum after an upset win two weeks ago at Chelsea, which rebounded with a 4-0 win at Steaua Bucharest.
The match was delayed by a protest against Russian gas supplier Gazprom which is the shirt sponsor of Schalke and a title sponsor of the Champions League.
The activists, apparently supporting environmental group Greenpeace, descended in the sixth minute and unfurled a banner which read ''Gazprom. Don't Foul The Arctic.''
When play resumed at St. Jakob Park, slack defending by Basel almost led to a goal.
Fabian Schaer miscued an attempted clearance and the looping ball dropped into the path of Schalke midfielder Kevin-Prince Boateng, who placed his shot just wide.
Basel's highly-rated Egyptian winger Mohamed Salah was closely monitored but eluded four defenders surrounding him to fire a rising shot wide in the 19th.
The Swiss champion threatened in the extended first-half stoppage time when veteran forward Marco Streller sent a glancing header just wide as Schalke goalkeeper Timo Hildebrand scrambled across his goalmouth.
Schalke threatened in the 52nd when goalkeeper Yann Sommer struggled with a cross to the far post, and a poor clearing header teed up Neustaedter for a powerful shot which struck the crossbar.
Draxler scored the clincher two minutes later when left unmarked to set himself and shoot after a corner was cleared weakly.
Basel failed to put Schalke under any sustained pressure, though Hildebrand dived to his right in the 76th to palm away a sweeping left-foot shot by substitute Giovanni Sio.
Basel now faces disciplinary action from UEFA - whose President Michel Platini attended the match - because host clubs are responsible for security at stadiums and ensuring matches run smoothly.
''The incident will be notified in the official reports of the UEFA delegate and of the referee, which will be sent to the UEFA administration for review and the potential opening of disciplinary proceedings,'' the European football governing body said in a statement.
UEFA has been asked by the Switzerland Football Association to consider Basel as a candidate to host matches at the 2020 European Championship, which is to be staged in 13 countries.