The Reds have not picked up any silverware since lifting the 2006 FA Cup and have not won the domestic title since 1990.
Last season also saw Liverpool lose their grip on the title record after fierce rivals Manchester United set a new benchmark of 19 top-flight crowns.
Carragher is aware the Reds can not hold on to records if they are not competing at the highest level and feels it is too long since the club enjoyed success.
"You want to be in there, and Liverpool should be in there challenging for trophies," Carragher told Sky Sports News.
"It's been a few years now since we won a trophy and that's what we want to try and change this season coming, but I'm sure everyone's got the same idea."
He added: "If Man Utd win, they win and you say good luck and you have to respect what they've achieved.
"We can't keep these records - we've five Champions League titles - if we're not competing for them.
"We have to try and win more as it's been too long for Liverpool and it's something we're looking to change."