Heartbroken Nina, 19, called the BBC panel show "extremely insensitive" after they debated whether convicted criminals should be allowed to move across Europe's borders.
14-year-old Alice Gross's body was found in the River Brent in West London, causing police to increase search efforts in the hunt for convicted murderer Arnis Zalkalns.
The 41-year-old builder, who was jailed for seven years in Latvia over the murder of his wife, is now being treated as prime suspect after he was spotted 'following' Alice in CCTV footage shortly before she disappeared.
He went missing a short time after Alice disappeared, and has not been seen since.
Although Alice’s name was not mentioned on BBC Question Time, the question: “Does Freedom of Movement across EU borders include convicted criminals?” was clearly sparked by the tragedy.
Alice's sister Nina responded: "It is extremely insensitive to use my families tragedy for political agendas and discussion.
LANDLADY OF ALICE GROSS SUSPECT REVEALS HIS SHOCKING MURDER CONFESSION
"This is a time of grief for our family."
She added: "This is a personal tragedy which we want to deal with privately, rather than fearing anyone using it to for any political agenda.
"Now is not the time for these discussions; it is exceptionally insensitive to talk about these issues at this time."
The BBC posted an apology to Nina and the rest of Alice's family, writing: "Dear Nina, we're sorry to hear this. We're really sorry for any hurt or offence caused by tonight's programme"
Meanwhile, after police confirmed on Tuesday that the body was that of Alice, her parents have issued a heartbreaking statement.
Rosalind Hodgkiss and Jose Gross said: "We have been left completely devastated by the recent developments and it is difficult to comprehend that our sweet and beautiful daughter was the victim of a terrible crime.
"Why anyone would want to hurt her is something that we are struggling to come to terms with. Alice was a loving and much loved daughter and sister, a quirky live spark of a girl, beautiful inside and out."
The family also urged people to help the police by bringing her killer to justice.