Friends and family of England players were showered with beer mugs in the stands after England drew 0-0 with Slovenia on Tuesday.
England had already qualified from Group C before kick-off, but fans have criticised Southgate for his disappointing performance.
Defender Ezri Konsa said his family was attacked “from all angles” and covered in beer. Family members were not targeted and the plastic cup was said to have been aimed at manager Gareth Southgate.
“I knew because some of my family members have been drunk,” Consa said.
The issue was not discussed between Southgate and the players after the Cologne game.
“We check on them. There are children in the stand with their families so we always have to check on them, reassure them and ask if they are okay.
“That happens in football.”
Fans across the English side of the stadium took home the trophy after Southgate’s side failed to score for 90 minutes.
Aston Villa’s Konsa says it’s “come from all angles” after his brother suffered an injury.
“We try not to think about it too much so we move on.”
“I understand what they’re telling me, it’s better than talking about the team,” Southgate said in his post-match press conference.
“This creates an unusual environment to operate in. I’ve never seen another team qualify and receive similar (treatment).”
England top Group C with five points and will face Slovakia in the last 16 on Sunday.
