"He felt well in the finale but we are taking him for an X-ray just to check his wrist," Brailsford told reporters, adding that updates would be posted on the team website.
Froome also sustained bruises but Team Sky sports director Nicolas Portal, who talked to the British rider during the stage, said he was OK.
Froome fell off his bike five kilometres into the 163.5-km ride from Le Touquet but quickly got back after bruising his left hip, shoulder, knee and elbow.
"It's alright, it's alright. We got really scared because it was a heavy fall but he seems to be OK. He says it's OK," said Portal.
Froome, fifth overall and two seconds behind Italian Vincenzo Nibali, crashed on the same side during last month's Criterium du Dauphine.
After Tuesday's incident, he received treatment from race doctor Florence Pommerie and was being helped back into the peloton by his Sky team mates.
"He's got bruises on his left side. He must be hurting a little," said Pommerie.
"His wrist is also hurting," she added.
Portal handed over a splint to Vasil Kiryienka for the Belarussian to bring it to Froome, whose wrist will be put to the test on the cobblestones that mark Wednesday's 155.5km fifth stage from Ypres to Arenberg Porte du Hainaut.
"We're taking as much precautions as we can," said Portal.
Another overall contender, Dutchman Baike Mollema, was caught up in the crash, but did not seem injured.
Alarm bells have been ringing in the months leading up to the race with Froome suffering from a back injury and then crashing at the recent Criterium du Dauphine.
(Reporting by Julien Pretot; Editing by Amlan Chakraborty/Tony Goodson/Alan Baldwin)
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