The 26-year-old has never been far away from the headlines since swapping Ajax for Anfield in January 2011.
The Uruguayan forward was handed a hefty ban after being found guilty of racially abusing Manchester United left-back Patrice Evra and on Wednesday night made his first appearance after a 10-match suspension for biting Chelsea's Branislav Ivanovic.
"Of course, any types of incident of that nature are damaging to a brand, but Liverpool Football Club is way, way bigger and always will be way bigger than any one player or one individual," Ayre said at the Nolan Partners Sport Industry Breakfast Club.
"What is important at those times is to ensure that we act respectfully and professionally as a football club.
"In the past we got some of that wrong and I'd say more recently we got that right.
"Luis is a footballer, sort of a street fighter, a larger than life character and we have seen that across not just even the Premier League but across football and other sports.
"Nobody is condoning any bad behaviour, but it is just something you have to deal with.
"He is what he is, is the character he is, and we have to try and harness that."
Suarez impressed in Liverpool's 1-0 Capital One Cup loss at Manchester United - a display few had expected to see after he had appeared determined to leave during the summer.
Principal owner John Henry waded into the matter by insisting the player would not be sold at any price and in any case Ayre has been impressed by the player's positive response to matters that have impacted the club as a whole.
"We have worked a lot with Luis since the last incident with Ivanovic and he has responded well to that," he said.
"He has been prepared to commit to that sort of work on his character.
"We saw him back last night [Wednesday] and he looks like the good bits of Luis, which is terrorising defenders and we're pleased he is back, pleased he is contributing.
"As we said all along through the summer, we expected him to remain a Liverpool player. He has remained a Liverpool player and we all move on together.
"I saw some of his comments this morning [Thursday] that he was there for the team and wants to help us achieve whatever we can achieve this season."
In those comments, in the aftermath of a committed performance on his return, Suarez pledged to do all he could to help Liverpool climb the table.
That was in contrast to his attitude early last month when, in the wake of a £40,000,001 bid from Arsenal - which they thought would trigger a release clause - he claimed manager Brendan Rodgers and the club had reneged on a deal to allow him to leave for a side competing in the Champions League.