Draxler's goal came in the second half of a game which earlier had to be halted after only four minutes when Greenpeace protesters abseiled down from the roof of Basel's St Jakob-Park stadium.
Spanish referee Alberto Mallenco halted the game after the protesters unfurled a banner which contained a message against Schalke sponsors Gazprom.
The banner was removed and Mallenco resumed proceedings following a short delay. The protesters were removed from the roof and were arrested by police.
UEFA later confirmed that disciplinary action might be taken against Basel over the protest but only after the match delegate's report has been submitted.
Schalke's Kevin-Prince Boateng had the first opening after 10 minutes but his shot was off target.
The visitors threatened again but when Atsuto Uchida crossed Max Meyer's volley from the edge of the area was blocked by Ivan Ivanov.
Draxler was off target for Schalke after bringing down a cross and making room for his shot. Schalke continued to dominate the first half and with 30 minutes gone could have threatened again but for some poor control in front of goal.
A fine move by the away side split the Basel defence but poor control by Jefferson Farfan saw the chance go begging.
As Basel began to come more into the game as half-time approached Schalke picked up two yellow cards in quick succession.
Dennis Aogo was the first to be booked for a foul on Mohamed Salah and then Farfan was cautioned by referee Mallenco for his foul on Taulant Xhaka.
Five minutes before half-time Basel forced a rare corner but an inswinging delivery from the left by Marcelo Diaz to near post was cleared by Uchida.
Schalke suffered a blow just before half-time as Farfan limped off injured and was replaced by Adam Szalai.
Basel's Ivanov tried a speculative shot from distance but his effort was off target.
Draxler crossed for Szalai but his header was saved by Yann Sommer and then Roman Neustadter's volley hit the bar.
Schalke pressed their superiority and took the lead after 53 minutes when Draxler fired a 20-yard volley into the top right corner of the goal.
Schalke's Timo Hildebrand saved his side when he beat away Giovanni Sio's shot for a corner and then punched away Marcelo Diaz's flag kick.
Draxler tried to add a second late on but the visitors had already done enough to secure all three points and go top of their group.