The subject of player behaviour has again been making headlines after it emerged Bairstow greeted Cameron Bancroft with what the Australian opener described as a headbutt in a Perth bar on October 29.
It follows on from all-rounder Ben Stokes remaining in the UK as he waits to discover if he will be charged after a late-night incident in Bristol in September.
England and Wales Cricket Board (ECB) director Strauss has spoken to Bairstow about the Perth incident, and while he will address the players again in Brisbane on Monday, he is satisfied no disciplinary measures will be appropriate this time.
"These guys are not thugs," he said.
"These are good, honest, hard-working cricketers who sacrifice a lot to play for England. They have done some great things in an England shirt, and I will back them on that to the hilt - because I know them.
"The guys were out for a couple of drinks, there was no curfew in place [and] ]they were free to enjoy an evening out."
Bairstow, however, does appear to have committed an error in judgement.
"Jonny Bairstow mentioned to me that he 'bumped' Cameron Bancroft," added Strauss.
"It's a greeting he does with his mates - and he assured me there was no malice, no intent, no aggression in what he did.
"Although I'm somewhat surprised he would choose to do such a thing, I'm taking him at his word. As such, I don't think it would be appropriate for us to be launching disciplinary proceedings against him.
"I have clearly reminded him of his responsibilities as an England cricketer - and I think he has obviously seen first-hand how in an Ashes series, with a lot of noise and attention and interest, small issues can become bigger issues."
Bancroft, who marked his debut with an unbeaten second-innings half-century as Australia coasted to victory in the opening Ashes Test at the Gabba, described Bairstow's introductory gesture as "weird".
Strauss added: "We have done a lot of work over the last five or six weeks, making sure the players understand their responsibilities and obligations and coming up with a code of conduct for them.
"I think we also need to understand and recognise that sport is moving on.
"What might have been acceptable in the old days is no longer acceptable, and we as an England cricket team ... need to move along with that.
"I think the players needs to be smarter. That's the reality - they are adults, intelligent adults, and at times they are not using that intelligence in the right way."
"He is shocked," Strauss said of Bairstow.
"He is surprised this has been made a big issue, but I think also he is quite contrite and realises what he thought was a bit of fun at the time can in the build-up to an Ashes series be construed as something completely different."