TDs fear the level of personal abuse they receive will escalate when the contentious abortion issue comes to a head in the new year.
In the wake of the death of junior minister Shane McEntee, politicians are calling for a clampdown on anonymous cyber-attacks and some calm in the debate.
Abuse on social media websites is believed to have been a contributory factor in Mr McEntee's death.
Thousands are expected at the Junior Agriculture Minister's funeral at the Church of John the Baptist in Nobber, Co Meath, today.
Due to the size of the crowds, mourners had to be ferried by bus to the removal last night of the Meath East TD, who took his own life on Friday.
His wife Kathleen was supported by their children, with Mr McEntee's tearful mother Madge following behind.
A framed photograph of Mr McEntee and his family was placed on the coffin, and Nobber parish priest Fr Seamus Houlihan said nobody but God could say what led the popular 56-year-old to take his life.
"Shane died on the shortest day of the year," Fr Houlihan said. " The darkness had almost overcome the world, and God knows because he didn't know and neither did the rest of us what darkness or what issues were working inside him."
TDs believe his death must be a "wake-up call for everybody" as they are concerned about the level of vitriol directed at politicians.
Former Taoiseach John Bruton, a friend of Mr McEntee, who replaced the former Taoiseach as Fine Gael TD for Meath, called for an end to the practice of messages from anonymous people being read out on TV and radio.
A TD who recently had to report a death threat to gardai wants to see political parties taking the lead in toning down the vitriol on the internet.
Fine Gael TD Jerry Buttimer said since the Budget he has been sent death threats and sinister comments about his family members.
He had to report some of the messages to gardai, including a text to his phone that read: "Have your family ready for a funeral."
"I'll argue the point with people on their views of policy, but the level of abuse since the Budget has been horrendous," he said.
The chairman of the Dail's health committee said political parties needed to take the lead.
He said those sending abusive messages were sometimes members of political parties.
"Political parties need to tell members to tone down on the websites. Members of the media have got a role to play too. Most of it is responsible, but some of the journalism is sensationalist," he said.
Mr Buttimer said the worst comments come in online chatrooms where people do not have to identify themselves.
"Some of the stuff posted on these websites is vile. If you were to read some of the stuff on those websites about you, you wouldn't get out of bed."
Following on from Mr Kenny's comments, Mr Buttimer said TDs were concerned about the abortion debate being made personal.
Mr Bruton said he wanted to see an end to anonymous messages being read out on radio talk shows.
"There has been an increase in this, particularly thanks to the anonymity that people enjoy in tweeting in comments, which are then broadcast and given status in radio programmes, as if they were coming from someone prepared to stand over what they were saying. In fact, they are being delivered under the cowardly cloak of anonymity."
But Green Party TD Dan Boyle said he couldn't take Mr Bruton's comments on anonymous texting seriously.
"Fine Gael has long run campaigns of 'ordinary' people texting," he said, a reference to the party encouraging members to contact radio stations.
Last week, Mr Kenny said he had received a message threatening revenge for his criticisms of the Vatican in the wake of the Cloyne Report.