Platini said UEFA has empowered referees to "temporarily stop the game and finally cancel the game if this racism keeps rearing its head."
Concern was fueled by a British television program last week showing discrimination and violent incidents at recent club matches in the co-host nations.
Ahead of the Euro 2012 kick off on Friday, the head of a Europe-wide campaign against discrimination in football acknowledged it posed more potential problems than previous tournaments.
"There is no question we are worried about this tournament more than any other," Football Against Racism in Europe (FARE) director Piara Powar told reporters on the sidelines of Platini's meeting.
UEFA invited the FARE network to help select a group of 31 expert spotters who will monitor every team's fans for any banners, chants and behaviour inside stadiums which breach UEFA's zero-tolerance anti-discrimination policy.
Powar said that Platini "understands what is going on," and that a team could even be expelled from the tournament for a third conviction from UEFA's independent judicial bodies.
Four years ago, UEFA fined the Croatia national association 20,000 Swiss francs (then $19,600; €12,450) for its fans neo-Nazi flags and chants during a Euro 2008 quarterfinals loss against Turkey in Vienna, Austria. At Euro 2012, Croatia will play Spain, Italy and Ireland in a group based in Poland.
Platini said he had not seen the BBC documentary, titled "Stadiums of Hate," and initially suggested that Poland and Ukraine offered a soft target for criticism days before they hosted the tournament.
However, he acknowledged that racism was a problem for society across Europe - including in England and his home country, France.
"There is more and more nationalism in Europe. You can feel this at a number of football matches," Platini said in French. "There are some worries, some big worries, but a lot has been done thanks to football."
Platini said UEFA would "always" support the decisions of its referees, who were given authority three years ago to respond to hostile incidents by halting matches.
"We will stop the game if there are problems because I think racism is the worst of this," he said.
However, Platini warned Italy forward Mario Balotelli that he would get a yellow card for fulfilling a recent threat to walk off the field to protest any abuse.
"It's not the player, Mr. Balotelli, who is in charge" of refereeing a match, the UEFA president said.
UEFA referees director Pierluigi Collina said later that the 12 match officials at Euro 2012 had briefed the 16 competing teams on standards expected at the tournament.
"Referees have a protocol so they know what they have to do. The referees are ready," Collina said.
Platini also downplayed questions relating to Ukraine's record on human rights and its jailing of former Prime Minister Yulia Tymoshenko for abuse of office.
The government in Platini's home country has said its officials will boycott matches in Ukraine - where Les Bleus will play - to protest the treatment of Tymoshenko.
"UEFA does not get involved in politics," Platini insisted. "Western countries can boycott if they want to. It's my role to simply organize the competition."