Ajax never got dragged through the mud by Suarez. Liverpool's reputation got spoiled because of their ridiculous and shameful behavior during the racism saga. They made declarations of his innocence based on his words alone, had their players and DoF give conflicting and suspiscious comments and made a mockery of the FA verdict where they did not appeal, but, still criticized the decision. And the Kenny Dalglish and team t-shirt thing was probably the worst PR decision taken by any English team in the Premier League era!
But for their stupidity, they wouldn't have been intrinsically associated with Suarez's misdemeanors.
Look at how publicly they're dealing with the transfer saga now. There is a tendency in them to be overtly public in all of their actions. Perhaps, they do it in order to keep the most blinkered/deluded section of their fan-base (which also happens to the most loud and activism-prone) in their side. But, they certainly don't have the charisma and the sunniness to pull it off and end up becoming a joke to the rest of the world whilst satiating their "online ultras".
Like most other clubs in the world, we'll be much more effective than them in dissociating our image from an errant star player's at any point of time in the near future.