Last week, FIFA ordered Hungary to play two home matches behind closed doors, one suspended for two years, and handed out a 200,000 Swiss francs (over £158,000) fine.
It followed charges of racist behaviour during the 4-0 World Cup qualifying defeat by the Three Lions in Budapest earlier this month.
Then on Thursday, the same day that Southgate named his 23-man squad for the qualification double-header against Andorra and at home to Hungary - a man was handed an eight-week prison sentencing over the online abuse of West Brom midfielder Romaine Sawyers.
Southgate refused to criticise Hungary ahead of the game on September 3, citing the racist abuse aimed at Bukayo Saka, Marcus Rashford and Jadon Sancho following their penalty shoot-out misses in the Euro 2020 final defeat to Italy as proof that there are similar problems at home.
Asked if he welcomed the harder action taken with recent charges and whether he thought Hungary's punishment was severe enough, he told BBC Radio 5 Live: "Well, I think it's very difficult to assess what is the right level of punishments or admonishments for things.
"Where we've come from as a team over the last few years is we think it's important to raise awareness of the issues, we've tried to behave responsibly around that and we do feel as if we're heading towards a more tolerant society and people are recognising that these things are unacceptable.
"So we could debate, a level of fine or a ban for hours and hours and very difficult to judge that but I think the biggest factor is that action is being taken and I feel like as though we're heading towards a more tolerant society."
West Brom supporter Simon Silwood, 50, was jailed and ordered to pay £500 in compensation over a "grossly offensive" Facebook post. Silwood was arrested after posting a message on social media saying Sawyers should win the 'Baboon d'Or' - a callous reference to the Ballon d'Or trophy - following his team's 5-0 defeat to Manchester City at the Hawthorns on January 26.
He denied the offence, blaming predictive text and telling police his phone had autocorrected the word "buffoon" to "baboon".
Silwood, who has been banned for life by West Brom, was convicted at an earlier hearing at Walsall Magistrates' Court after District Judge Briony Clarke ruled he was "not a credible witness" and had meant the post to be offensive.
"I've only just learned about the specifics of the case," added Southgate. "Look, I think all I would say is that it's important that we do take the online abuse, the abuse in general society, very seriously. I think we're recognising the importance of that.
"We want a tolerant society and I think that's where we're heading. For me, they're the important messages at this time."