"The court's ruling in this matter is as tainted as the original verdict is unsafe."
The statement said that the issue over Juror No 50's revelations to media outlets after the trial will be one among many issues that will be appealed to the 2nd US Circuit Court of Appeals in Manhattan.
"Our family is optimistic about Ghislaine's success on appeal," they wrote.
Juror No 50's media interviews days after the verdict came after a month-long trial at which Maxwell was portrayed as the crucial component of Epstein's sexual abuse conspiracy. Sometimes, prosecutors said, Maxwell joined in the abuse.
After the trial's conclusion, the juror, identified in court papers only as Juror No 50, said publicly that he'd been abused as a child and had persuaded some fellow jurors that a victim's imperfect memory of abuse doesn't mean it didn't happen.
He was among potential jurors in the case who filled out a 50-page questionnaire including a question that asked: "Have you or a friend or family member ever been the victim of sexual harassment, sexual abuse, or sexual assault?"
The juror checked "No".
The juror said in one interview that he didn't remember being asked that question, which was No 48 on the form.
After rejecting defence demands that she immediately order a new trial, the judge conducted an unusual hearing at which she questioned the juror.
His answers were analysed by the judge in her written decision last Friday.
She said she concluded that the juror's rush to finish the questionnaire led to inaccurate answers.
"Juror 50's lack of attention and care in responding accurately to every question on the questionnaire is regrettable, but the court is confident that the failure to disclose was not deliberate," she wrote.
"The court further finds that Juror 50 was not biased and would not have been stricken for cause even if he had answered each question on the questionnaire accurately."
Defence lawyers had insisted that they would have used one of a limited number of "strikes" that allow them to eliminate several jurors for any reason at all to force him off the jury.
In its statement, Maxwell family members said the judge had failed to uphold "the paramount interests of justice" by severely limiting the questioning of the juror and had "effectively ensured the loading of the dice" by granting him immunity from prosecution for perjury.
Epstein was 66 when he committed suicide in 2019 as he awaited a sex trafficking trial in Manhattan.