The Premier League said in June it will introduce enhanced safety measures to prevent crowd misbehaviour, with clubs unanimously agreeing that visiting teams will provide additional stewards for away matches.
Dier said some of his family and friends faced problems at Stamford Bridge during Tottenham's 2-2 draw with Chelsea in the league last month.
"For me it [fan behaviour] is a serious problem. I had some family and friends at the Chelsea away game with Tottenham and they had problems and stuff," Dier said.
"I wanted to emphasise it was both sets of fans -- I am not saying it is Chelsea fans or Tottenham fans, it is football fans in general."
In 2020, Dier was banned for four games and fined by the English Football Association over a confrontation with a fan after Spurs' defeat by Norwich City in the FA Cup. British media reported the supporter had argued with his brother.
"We played Burnley after I went in the stands for the next away game and the Burnley fans were singing a song about my brother and I like that kind of thing -- I find it quite funny," Dier said.
"But there are some things I find very strange. It is not nice. My family would never go to an away game nowadays because of it -- and that's a shame that I feel too uncomfortable for them to go to away games."