Reds striker Mario Balotelli was charged on Friday by the Football Association for an Instagram post.
The 24-year-old reposted an image which appeared to contain anti-Semitic and racist references before apologising.
"Given the choice we'd love to keep all our players away from social media because it really is a problem," Ayre told BBC Radio 5 live's Sportsweek.
"I can honestly say there isn't any more we could do as a club to try to educate our players around this but it still rears its head at least once a season - probably at every club in some form or another."
The picture posted by Mario Balotelli on Instagram Balotelli forwarded on a picture produced by someone else
"One of the problems that all footballers have had for many a year is that they have too much time to kill," added Ayre.
"In that downtime they seem to put these crazy sort of messages up. It is definitely an issue.
"I don't think we have got to the point where we would stop people doing it but we certainly have a very stringent policy that all of our players sign.
"If they step out of line then we punish them."
Balotelli is currently side-lined with a groin injury and has missed Liverpool's last five games. He has been a huge disappointment since signing for Liverpool in the summer and rumours are rife that Liverpool might try and cut their losses in January.