"It was not being used to enhance performance," Brailsford told the BBC, adding that Team Sky "do not cross the line" when it comes to performance-enhancing drugs.
Brailsford said Sky would seek to be more transparent in future about riders' use of Therapeutic Use Exemptions (TUEs).
Wiggins has faced questions over the timing of the medical interventions and his use of a corticosteroid to tackle asthma problems. Details were published by cyber hackers on the fancybear.net website this month.
The data revealed the Briton was given permission to take triamcinolone before the 2011 and 2012 Tours as well as the 2013 Giro d'Italia.
TUEs allow athletes to take banned substances for verified medical needs and are signed off by sports federations. There is no suggestion Wiggins broke any rules.
"I have known Bradley a long time and he is an asthma sufferer and he has struggled with allergies for as long as I have known him," Brailsford said.
"I know that at the time there was a recommendation to see a specialist, he went to see a specialist and was then given permission by the authorities.
"I trust and believe in the integrity of that process."
Brailsford added that he had been aware at the time about what Wiggins was taking and had complete trust in the team doctors.
He did not believe the reputations of rider or team, who have a much-publicised 'zero tolerance' policy, had been tainted.
Wiggins told the BBC in an interview broadcast on Sunday that the TUEs were "to cure a medical condition.
"This wasn't about trying to find a way to gain an unfair advantage. This was about putting myself back on a level playing field in order to compete at the highest level," he said.
The injections of triamcinolone have been highlighted as an apparent contradiction of Wiggins' previous claims that he had adhered to cycling's "no needles" policy.
A spokesman for Wiggins said last week that the injection mentioned in the leaked information was an intramuscular treatment. He said the rider stood by his comment concerning the use of illegal intravenous needle injections.
Questions have also been asked, in a sport with a notorious doping past, as to why Wiggins did not need the injections before 2011 or after 2013.
"In our total existence we (Team Sky) have had 13 TUEs," said Brailsford. "This is not a systematic abuse of TUEs, we have raced many, many races and there's a very small number. We run a tight ship."