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Protests broke out over labour law reforms in May, while 130 people were killed in attacks on Paris in November.
Desailly, 47, said there should be no fears over security, adding that his "own kids are going to the fanzone".
"We need football. We need football to be there for us to enjoy and to have hope," he told BBC Sport.
"France is going through a little bit of trouble. I'm sure it's going to be secure. The information I have, people will be protected."
Desailly was part of the France squad that won the 1998 World Cup on home turf and Euro 2000 in Belgium and the Netherlands.
The former Chelsea defender said Atletico Madrid forward Antoine Griezmann will be "the man of the Euro".
"We need him because he has shown that he can score," said Desailly, who also expects West Ham's Dimitri Payet, Manchester United's Anthony Martial and Arsenal's Olivier Giroud to impress.
"I am hoping that Payet or Martial will play in this offensive formation. But we need Giroud for Griezmann to express himself."
Desailly, who spent five years in Serie A with AC Milan, said Juventus midfielder Paul Pogba is "one of the major players with a problem".
"He has so much quality but is he an offensive player or a defensive player? We need him to protect the defensive block," he said.
"Pogba should concentrate on one thing, not trying to be everything - being a playmaker, being a defender, being a skilful player."
Desailly thinks France have a chance of winning the tournament if they can "wash away all this pressure", but said England "will surprise everyone".
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France needs Euro 2016 to help repair the nation after recent economic and security troubles, says World Cup winner Marcel Desailly.