The co-hosts reach a milestone on Sunday when the draw for the qualifying competition is held at Warsaw's Palace of Culture and Science at 11am GMT.
Platini, speaking at a news conference, said: "There is a long list of things that need to be improved, but we are strictly dealing with minor problems.
"We have many minor issues. Enormous efforts have been undertaken by Poland and Ukraine and their football federations.
"We have got two years to go to the finals, but to be frank we are not dealing with major problems at the moment. Everything is as it should be."
Previously, problems with stadium construction, infrastructure, communications and accommodation caused Platini to warn the hosts they risked losing the right to stage the event, which is due to take place in June and July 2012 in four cities in each country.
Warsaw, Gdansk, Poznan and Wroclaw will stage matches in Poland with Kiev, Lviv, Kharkiv and Donetsk the four Ukrainian host venues.
Platini said one of the issues still to be resolved was which Ukraine cities would stage the quarter-finals, but he added: "We have two years to decide that, it is not a major worry."
‘Slot hasn’t come in and changed everything, making it all about himself – it’s been about the players he’s inherited, getting the best out of a system he likes’: Liverpool manager impressing former players with personality and tactical approach
‘We thought we’d signed him, but Chelsea blew us away. I think they signed him so we couldn’t have him, as much as because they really wanted him’: Liverpool actually failed to sign Mohamed Salah in 2014, three years before finally getting the Egyptian
