The friends and families of England players were hit by beer cups in the stands after England's 0-0 draw with Slovenia on Tuesday.
England had already qualified from Group C ahead of the game but fans rounded on Southgate following a disappointing performance.
Defender Ezri Konsa said the families were hit "from all angles," getting covered in beer. It is reported that the families were not targeted and the plastic cups were aimed at manager Gareth Southgate.
"We were aware of it because I think some of our family members got hit with a few drinks." Konsa said.
The matter has not been discussed between Southgate and the players after the match in Cologne.
"We check on them. There's kids in the stands with family so we always have to check on them, reassure them and ask if they're all right.
"In football stuff like that happens."
The cups had come from fans all around the English end of the stadium after Southgate's side failed to score in 90 minutes.
"It was coming from all angles," Aston Villa's Konsa said after his brother got hit.
"We try not to think about it too much so we move on."
Southgate said in a postmatch news conference: "I understand the narrative towards me and that's better for the team than it being towards them.
"It is creating an unusual environment to operate in, I've not seen any other team qualify and receive similar [treatment]."
England topped Group C with five points and will face Slovakia in the round of 16 on Sunday.