Karius made mistakes for two of Real Madrid's three goals and has been sent hundreds of threatening messages on social media.
Merseyside police are investigating, and a spokeswoman said they are taking the messages "extremely seriously and any offences identified will be investigated," per Wil Crisp at The Telegraph.
One message said his "sons deserve to die of cancer" and another read "I hope your whole family dies," per the report.
Real Madrid took the lead in the second half when Karius attempted to throw the ball to a team-mate but it was intercepted by striker Karim Benzema who scored.
Television pundit Gary Lineker said it was a huge mistake:
Sadio Mane equalised for Liverpool before Gareth Bale scored a brilliant overhead kick to restore Real Madrid's lead. However, the Welshman then scored again to put the game out of Liverpool's reach after Karius fumbled his shot from range.
Henry Winter at The Times offered his view of the incident:
Karius was in tears at the end of the game and went over to the Liverpool supporters with his hands raised in apology.
He also offered an apology via social media:
The goalkeeper was consoled by Real Madrid players after the whistle, per football presenter Mark Pougatch:
There is no doubt that Karius made mistakes that ultimately cost Liverpool the game. Karius has a reputation for errors, but he has improved this season and had been made the team's first choice goalkeeper ahead of Simon Mignolet by manager Jurgen Klopp.
However, he is not viewed as one of Europe's best goalkeepers, and long before the game, there was a sense Liverpool needed to buy a replacement.
Roma President James Pallotta has said Liverpool want to buy goalkeeper Alisson, per Sky Sport Italia (h/t Rhys Turrell at the Daily Star).
Karius' performance in Kiev will have been traumatic for the goalkeeper, and it also raises further questions over his future at Liverpool.
Former Reds defender Mark Lawrenson said he does not "see any way back for him at Liverpool now," per BBC Sport.
Yet Karius deserves sympathy, not abuse, after failing so spectacularly in what was the biggest moment of his career. The goalkeeper has admitted his mistakes and vowed to return stronger, but whether he gets the chance to do that at Liverpool remains to be seen.