Manchester City midfielder Toure was subjected to monkey chants during the Sky Blues' Champions League win at CSKA Moscow two weeks ago, and subsequently suggested black players may boycott the 2018 World Cup if Russia does not do more to tackle the problem.
UEFA's punishment of CSKA was to demand a partial closure of their stadium for their Champions League meeting with European champions Bayern Munich later this month.
Swansea head to Russia as they look to maintain their lead at the top of Group A, making them the first Premier League side to visit the country since the incident involving Toure, and winger Dyer admits recent events have played on his mind.
He said: "You are always fearful when you come to a place that has a reputation, but there's nothing you can do about it. If it happens, it happens. You have to follow the correct steps and hope the authorities deal with it.
"You don't know how you would react in the heat of the moment. But hopefully it doesn't come to that - hopefully it's a good game of football and the crowd are fine."
He added: "It (racism) has been there forever and I can't really see it going away. We just have to make sure we keep punishing those that do it and hopefully it will quieten down ."
But the former Southampton man hopes Austrian referee Robert Schorgenhofer will take the correct steps in the event of any unsavoury behaviour, something which did not appear to take place when Toure complained to Romanian official Ovidiu Hategan.
UEFA guidelines to referees, when it comes to dealing with racist behaviour from supporters, state that if a player complains a message for fans to desist should be relayed within the stadium, this was not the case at the Manchester City game.
If that fails the official in charge can choose to temporarily halt the game for a further warning to be aired, with abandoning the game the final step.
"It's not nice what's going on. I feel it is the duty of the football authorities to address that," said Dyer.
"If things happen while we are in Russia we have to talk to the referee and he needs to take the correct steps. That's all we can do. We have to go about things the right way."
Dyer is happy to rely on the officials to deal with any issues, should they arise, but made it clear he felt UEFA's punishment of CSKA could, and should, have been stronger.
"I think the punishments need to be greater and then the clubs will take more responsibility for their fans," he said.
"They will want them to stop doing it because otherwise they will get punished, so they will be killing themselves.
"There definitely needs to be greater punishment for teams so they have to control their fans."