The Dutch forward linked to a mocked-up picture of referee Howard Webb in a Manchester United shirt after Liverpool lost 1-0 at Old Trafford in the FA Cup.
Babel has been defended over his message on the social networking site.
"I think it was a bit tongue in cheek," said Reds caretaker boss Kenny Dalglish, who is also a Twitter user.
The Scot accepted that Babel should be punished but does not believe he deserves a severe sanction.
"Ryan has to be responsible. If you are going to be on Twitter you have to be responsible," stated Dalglish.
"You know the repercussions if you say or do something which is contentious.
"He never drew it [the picture] up, he just forwarded it on.
"The club will take some action I would have thought and then that should be it for me."
In the FA Cup match on 9 January, Ryan Giggs scored the game's only goal from a second-minute penalty awarded by Webb for Daniel Agger's challenge on Dimitar Berbatov.
Webb later sent off Steven Gerrard in the 32nd minute for a challenge on Michael Carrick. Dalglish labelled the penalty decision "a joke" and also disagreed with the red card for his captain.
Among figures in the game who have defended 24-year-old Babel are Blackpool manager Ian Holloway.
"I thought it was funny - I thought the ref looked really cool in a Man United shirt to be honest," he commented last week.
"I saw him in the toilet at the match I went to the other night at Stockport. I said 'come on, what sort of game did you have against Man United? Was that a sending off and was that a penalty?'
"We had a bit of a laugh about it but to pick on young Ryan Babel for that - how can you fine him for that? It's rubbish." Gordon Taylor, chief executive of the Professional Footballers' Association, said the FA had been "too precious" in its reaction to Babel's tweet. "Where's our sense of humour gone in our game? He's apologised, let's move on," he stated.
In addition to posting a link to the doctored picture of Webb, Babel wrote: "And they call him one of the best referees. That's a joke." He subsequently removed the link and wrote an apology on Twitter.
The former Ajax star has more than 180,000 other Twitter users, including an extra 20,000 over the last week, following his updates and his messages - and the offending image was subsequently reproduced by some media outlets.
Babel later wrote on Twitter: "My apology if they take my posted picture seriously. This is just an emotional reaction after losing an important game. Sorry Howard Webb."
He then joked about spending a spell in "Twitter jail" and his previously prolific posts stopped on 9 January.
It is not the first time that a Liverpool member of staff has been drawn into controversy through their use of the website.
The club's head of fitness and conditioning Darren Burgess labelled England "completely amateurish" and "disgraceful" after Gerrard was injured in the 84th minute of an international friendly against France.
Babel's use of the site has also been questioned before, with the player revealing his displeasure at being dropped by then Reds boss Rafael Benitez in a tweet in January 2010.
Source: BBC Sport