Problematic live interviews can also have potential negative career consequences for those involved. Mr. Kelly and his wife immediately feared the worst, assuming that he wouldn’t be contacted again to appear on TV.

“We said to each other, ‘Wow, what just happened?’ ” Mr. Kelly, said, adding the blame was entirely on him for not locking the door.

He immediately wrote to the BBC to apologize, but within 15 minutes the broadcaster asked if it could put a clip of the interview on the internet. The couple initially declined, feeling uncomfortable that people might laugh at their children. But they were eventually persuaded that the video would show they were just a regular family.

Within a couple of hours, it became clear to them that the video would disrupt their lives. Mr. Kelly said his Twitter and Facebook notifications began going haywire as people shared the video online. The next day he put his phone in airplane mode as the number of emails and calls, many from journalists, became overwhelming.

The couple spent most of Saturday trying to decide how to handle the attention. Offers from major U.S. TV networks and media came flooding in. Some journalists tracked down Mr. Kelly’s parents in the east side of Cleveland to ask them about it.