"We have never been prouder of our son and brother Jim. He gave his life trying to expose the world to the suffering of the Syrian people,” the family wrote.
“We implore the kidnappers to spare the lives of the remaining hostages. Like Jim, they are innocents. They have no control over American government policy in Iraq, Syria or anywhere in the world," the statement said.
US President Barack Obama will make a statement on the matter later on Wednesday.
The video, captioned "A Message to America", was put on social networks on Tuesday night. There is no confirmation so far that the person shown in the video was Foley.
The journalist had been reported missing in Syria's north-west on November 22, 2012.