The men could face 16 years in jail if convicted
The family has asked the Irish Foreign Minister, Brian Cowen, to ensure that the men's lives are protected in prison.
Mr Connolly was charged along with Martin McCauley and James Monaghan with training Marxist rebles in Colombia on Wednesday.
The men had been held at a military base in Bogota under heavy guard since 11 August.
Two of the men are from the Republic of Ireland and one is from Northern Ireland.
'Innuendo'
Colombia's public prosecutor's office said the men would be held at an unspecified prison while the state prepared its case against them.
They are accused of training rebels from the Revolutionary Armed Forces of Colombia (FARC) to make bombs and other weapons, which carries a 16-year sentence.
Speaking on Wednesday, the Connolly family said they had endured 10 days of false accusations, wild allegations and innuendo aimed at determining the three men's guilt without any proper legal process.
The family also said they were particularly concerned that the men remained in custody in a country where there were daily reports of political murders and riots in prisons.
Meanwhile, the SDLP's agriculture minister, Brid Rogers has urged Sinn Fein to come clean over the three men's role in Colombia.
Sinn Fein has again denied the trio are party members.
However, the Ulster Unionist Party minister Sir Reg Empey said the government should take action.
"Republicans are engaged with drug-pushing Marxists in the jungles of South America," he said.
"An organisation that attacks American business, that has murdered and maimed, that is building itself up to carry itself into urban areas of Colombia and these people are running about with them," he said.
'Shocked'
The deputy leader of the Democratic Unionist Party Peter Robinson said the charges vindicated his party's decision to "expose the PIRA's activities in Colombia".
"The entire episode has exposed the folly of David Trimble and the Ulster Unionists who have appeased the representatives of terror and colluded in concessions to republicans," he said.
The Colombian Attorney General's office said it had enough evidence to proceed with the charge of providing training for Marxist guerillas from forensic tests and with inconsistencies in the Irish men's stories.
'False passports'
Under the Colombian legal system, prosecutors have 240 days to prepare the case before going to trial.
The men have also been charged with carrying false passports, to which they have already confessed and could be sentenced almost immediately.
Colombian Senator Juan Manuel Ospina is a member of the country's governing body.
Speaking on Wednesday, he said: "At the moment, certainly they are charged with having falsified papers."
The trio were detained as they attempted to leave the country.
The army said the training was carried out over five weeks while the three were in a demilitarised area the government handed over to the FARC more than two years ago to launch peace talks.
The incident has embarrassed the Colombian Government, which is trying to show its peace talks with the FARC are making headway.