The Premier League side began the group stages with a 1-1 draw away at Paris Saint-Germain but there was never any doubt that they would follow that result with three points in North London.
A dominant performance from the Gunners began as early as the seventh minute when Walcott netted in his third successive match for the first time since 2013 after he met Sanchez's cross to head home from close range.
Basel had their moments in the immediate aftermath of the goal but they could not force David Ospina into action, and they were eventually made to pay before the half-hour mark.
Again, Sanchez provided the assist for Walcott to score his second of the night with the pair playing a quick one-two before Walcott fired the ball in off the far post.
It soon became a case of how many goals would the home team score as they peppered the Basel backline but they were met by an inspired 15 minutes from goalkeeper Tomas Vaclik, who made numerous interventions to keep his side in the match.
Sanchez twice squandered openings inside the penalty area as Vaclik got himself in the way, while Mesut Ozil twice failed to hit the target from promising positions.
Arsenal wasted little time in trying to find a third goal when they returned for the second half but the sole chance of the opening stages fell to Alex Iwobi, who poked the ball wide of the post after a rampaging run towards goal.
Basel had offered very little in the final third but midfielder Birkir Bjarnason twice went close for the visitors, with his first shot being tipped over the crossbar by Ospina.
The second was a harder chance after the Arsenal stopper had failed to deal with a corner, but the Iceland international could only send a half-volley at the back post over the bar.
The goal sparked an improvement in Arsenal's play in the opposite half and they should have netted a third when Walcott reached the byline and crossed for Sanchez, but his teammate volleyed wide from six yards out.
As the game entered added-on time, Sanchez squandered another chance from inside the penalty area as Vaclik pulled off yet another block to deny the Chilean, but Arsene Wenger will be content with securing three points to match Paris Saint-Germain's victory over Ludogorets Razgrad.