The German made eight changes to the side that drew with Burnley in the Premier League at the weekend, including handing a first start to summer signing Alex Oxlade-Chamberlain.
The Reds dominated the first half at the King Power Stadium, but second-half goals from Shinji Okazaki and Islam Slimani consigned the visitors to a third-round exit from the Carabao Cup.
Sherwood criticised Klopp for not picking a stronger team against a fellow Premier League club and accused the German of not showing the competition enough respect.
"Certainly when you are playing against another Premier League club, you have to pick a team that you think can win the game," he told Sky Sports.
"And I think he has got it wrong tonight. Those Liverpool fans travelling home, they want to win a trophy. There are not many trophies out there to win and this is a good one to start off with.
"He has already been to a final and he needs to go one step further and nick a trophy, because at the moment it is not good for Liverpool Football Club.
"The pressure is mounting on him and this will not help tonight. It is alright being unlucky all the time, but people stop listening in the end. You need to get results."
The defeat against Leicester extended Liverpool's winless run in all competitions to four matches, with the Merseysiders already five points adrift of league leaders Manchester City.
With the club having only won one major trophy over the past decade, Sherwood said Liverpool should be taking cup competitions more seriously.
"They should show it more respect," he said. "There are not many trophies out there to go for. Winning the Premier League is very hard to do as people have seen. Winning the Champions League is even harder."