The family's lawyer Aamer Anwar said they wanted to express confidence in Police Scotland's new investigation.
He warned that this trust was "not unconditional" and that detectives from the original inquiry needed to answer for actions which had "betrayed Emma".
The family had been meeting the Lord Advocate James Wolffe at the Crown Office in Edinburgh.
Emma Caldwell, a 27-year-old heroin addict who had been working as a prostitute, was discovered dead in woods near Biggar in May 2005.
In May 2015, the Lord Advocate had told Police Scotland to reinvestigate her murder after the most senior officers in the Crown Office considered the case.
Reading a statement outside the Crown Office, Mr Anwar said the family had felt let down by the original investigation, which had failed to bring Emma's killer to justice.
He said: "The Caldwell family made it clear to the Lord Advocate that they had not forgotten that senior detectives from Strathclyde Police had betrayed Emma and they must answer for their actions one day."
He also said they had been frustrated about the length of time the new investigation was taking.
"The Lord Advocate tried to reassure the family that this is a painstaking complex investigation begun from scratch," he said.
Senior investigating officers told him that thousands of documents are being re-examined, which have generated numerous lines of inquiry that need to be investigated.
Police Scotland have also invited the Metropolitan Police to review its investigation strategy.
Mr Anwar said it had robust and painful meeting for the family, but they had welcomed "the dedication and commitment show by Police Scotland and the Lord Advocate to securing justice".
Making a direct appeal to the killer, he said: "Eleven years ago when you took Emma's life, you tore apart her family's lives forever.
"They were unable to bury Emma for some two years.
"Her mother Margaret has never been able to grieve and when William, Emma's father, died in 20001 he made his family promise they would never give up fighting for justice.
"Sadly it is inevitable that other women will have suffered at the hands of this killer and he will have aroused suspicions in his friends and family.
"The Caldwell family urges those who have such information to have the courage to come forward and speak to the police in total confidence."
After the meeting, a Crown Office spokesman said: "In May 2015 Crown Counsel instructed a re-investigation of the murder of Emma Caldwell.
"That investigation is complex and ongoing.
"The Lord Advocate today had the opportunity to meet with Emma Caldwell's family and to discuss the case with them."
The new investigation was launched after a BBC File on Four programme claimed that a local suspect was dropped in favour of prosecuting four Turkish men.
The programme revealed that officers who wanted to charge the local man, one of Emma's clients, were stopped by senior detectives investigating the Turkish suspects.
One of the Turkish men, who was charged with murdering Ms Caldwell, was awarded an out-of-court settlement after suing police for false arrest.
The family of Emma Caldwell who was murdered in 2005 have appealed directly to her killer to give themselves up.