The 26-year-old has received 19 games worth of suspensions since his £22.8 million move from Ajax in January 2011 without getting sent off, and Ayre acknowledged Suárez has harmed Liverpool's reputation.
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Suárez made his Liverpool return at Manchester United on Wednesday
"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 Suárez Controversies
(01 of18)
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Suárez stops the ball with the hands during the extra-time of Uruguay's 2010 World Cup quarter-final match with Ghana... (credit:Getty Images)
(02 of18)
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He is shown a red card after his handball by referee Olegario Benquerenca, but Asamoah Gyan misses the spot-kick... (credit:PA)
(03 of18)
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And Uruguay go on to win on penalties as Suárez is demonised by the Ghanians (credit:Getty Images)
Dutch Eredivisie - Bite incident Luis Suarez(04 of18)
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Suárez was not sent off for biting PSV Eindhoven midfielder Otman Bakkal's shoulder while Ajax captain in November 2011... (credit:Getty Images)
(05 of18)
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But he did receive a seven-match ban (credit:Getty Images)
(06 of18)
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An early glimpse of Suárez's nature came in March 2011 when, after a Rafael da Silva tackle on Lucas Leiva, he slyly tugged the Brazilian's hair in a bid to wind him up (credit:Getty Images)
(07 of18)
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Suárez feigns agony as Jack Rodwell is erroneously sent off by Martin Atkinson in the October 2011 Merseyside derby. Liverpool win the game 2-0 and Suárez scores. (credit:PA)
(08 of18)
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Later in October, a foul by Suárez on Patrice Evra sparks off his most infamous misdemeanour... (credit:Getty Images)
(09 of18)
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When he racially abuses Evra, calling him a "negro" eight times... (credit:Getty Images)
(10 of18)
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Suárez is fined £40,000 and banned for eight matches (credit:Getty Images)
(11 of18)
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Suárez is suspended for one game for making an offensive gesture to Fulham fans in December 2011.
(12 of18)
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On his first start since his ban for racially abusing Evra, Suárez refuses to shake his hand in February 2012 (credit:PA)
(13 of18)
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Suárez executes a pathetic dive in a 0-0 draw against Stoke in October 2012. He later admits to cheating (credit:AP)
(14 of18)
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Bemusing irony as Suárez opts for a diving celebration in front of Everton manager David Moyes, who accused the Uruguayan of simulation after the Stoke tumble (credit:PA)
(15 of18)
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Suárez is fortunate not to be red-carded after he rakes his studs down Sylvain Distin's heel in the same game. Everton fans enjoy poetic justice when he scores a late header which is wrongly disallowed for offside (credit:PA)
(16 of18)
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Suárez controls the ball with his hand at Mansfield Town to score Liverpool's winner in the third-round of the FA Cup, January 2013 (credit:AP)
(17 of18)
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Chewy Suárez: A shocked Branislav Ivanović reels from being bitten by the controversial striker (credit:Getty Images)
(18 of18)
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Giorgio Chiellini shows off the bite mark from where Suárez bit him. (credit:Getty Images)
"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."
Liverpool's response to Suárez's eight-match ban and £40,000 fine for racially abusing Evra was widely condemned. Then manager Kenny Dalglish and squad members wore T-shirts in defence of Suárez but Ayre said the club have worked to repair Suaárez's image following the Ivanović incident.
"We have worked a lot with Luis since the last incident with Ivanović and he has responded well to that," Ayre added.