The segment, which aired on the breakfast show on 10 August, saw reporter Simon Jones broadcast live from a boat that had pulled alongside a dinghy with migrants on board.
He said: “We have seen them trying to get water out of the boat, they’re doing that at the moment, they are using a plastic container to try to bail out the boat.
“Obviously it’s pretty overloaded there. People are wearing life jackets, it is pretty dangerous, just the number of people on board that boat.”
After asking whether they were safe, Jones remained with the boat until those on board were picked up by the UK Border Force.
Some 8,340 complaints were made to the broadcaster on the grounds that viewers felt the programme showed “offensive/insensitive coverage of migrants crossing the Channel by boat”.
In response, a BBC spokesperson said: “This report was a stark illustration of the significant risks some people are prepared to take to reach the UK.
“Channel crossings is a topic of huge importance and we always endeavour to cover the story sensitively.
“In this instance the Dover Coastguard were aware of the boat before our crew spoke to them and at no point did they, or those in the boat signal that a rescue operation was required.
“The Coastguards instead alerted Border Force, who then safely picked up the occupants and took them to shore.”
Earlier this month, the BBC received almost 19,000 complaints about the uncensored use of the N-word in a report about a racist attack in Bristol.