Wiggins, the 2012 Tour de France winner, has been in the spotlight over his past use of Therapeutic Use Exemptions (TUEs), which allow athletes to take banned substances for verified medical needs and are signed off by sports federations.
Brailsford, also a former British Cycling performance director, has denied any wrongdoing and said the TUEs were medically necessary.
"I'm not proud of the way I handled this. I relayed the information before I had the full facts. With hindsight, that was stupid, I've made it a damn sight worse than it needed to be," Brailsford told the Cycling Podcast.
"I gave a running commentary and on two occasions, that proved to be inaccurate. From what was a small fire, I've inadvertently thrown a huge amount of petrol on it.
"You've got to look yourself in the mirror, I've got to hold my hand up, I've not done a very good job of this one."
Brailsford was also pressed on the contents of the medical package but did not reveal what he knew of the incident.
Data leaked last month by the Russian-based Fancy Bears cyber hacking site claimed Wiggins had been given permission to have legal injections of the banned drug triamcinolone to treat breathing difficulties before the 2011 and 2012 Tour de France and 2013 Tour of Italy.
On each occasion, the TUE was approved by British authorities and cycling's governing body, the UCI, and there is no suggestion Wiggins broke any rules.
"I trusted the process and the system. There were no alarm bells ringing in my head. I think 100 per cent I'd have done the same thing again," Brailsford added.