"There were some regrettable incidents on the tour," Cookson told AFP in an interview at a International Olympic Committee meeting in Kuala Lumpur, Malaysia.
"I am worried about the beginnings of an element of hooliganism coming towards our sport which we have largely been able to avoid in recent years.
"Everyone needs to be a little bit careful for the future."
Cookson said the incidents could be "isolated" but had made him "very uncomfortable".
"I think Chris was subjected to a fairly nasty form of antagonism from a small number of people.
"But there was also a little bit of slightly joking, slightly hooligan-type behaviour in places with cars being kicked, things being thrown at cars."
Cookson said French authorities shared his concerns and warned that the freedom around cycling could be at risk.
"Over the years I have become more and more worried about people running alongside riders, pushing riders and so on and we are getting to a point where that's beginning to be a little more concerning.
"The Tour, all of cycle racing is a great free sport. It is possible to get close to the athletes, to the competitors, in a way that is not really possible in any other sport.
“If we want that to continue then we all have a responsibility to behave."
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