Friends and family of England players were pelted with beer glasses in the stands after England's 0-0 draw with Slovenia on Tuesday.
England had already qualified from Group C before the match, but fans turned against Southgate following their disappointing performance.
Defender Ezri Konsa said the families were hit “from all angles”, leaving them covered in beer. It was reported that the families were not the target and that the plastic cups were aimed at manager Gareth Southgate.
“We were aware of it because I think some of our family members were hit with some drinks,” Konsa said.
The matter was not discussed between Southgate and the players after the game in Cologne.
“We check on them. There are children in the stands with their families, so we always have to check on them, reassure them and ask them if they are okay.
“This kind of thing happens in football.”
Cups had come from fans across the English side of the stadium after Southgate's side failed to score in the 90 minutes.
“It was coming from all angles,” Aston Villa's Konsa said after his brother was hit.
“We try not to think about it too much and just move on.”
Southgate said in a post-match press conference: “I understand the narrative towards me and that's better for the team than it is towards them.
“It's creating an unusual environment to operate in, I haven't seen any other team qualify and receive a similar level. [treatment]”.
England topped Group C with five points and will face Slovakia in the round of 16 on Sunday.
