Campbell, the former Arsenal and Tottenham Hotspur defender who played 73 times for England and appeared at six major tournaments, told the BBC's Panorama programme: "Stay at home, watch it on TV. Don't even risk it... because you could end up coming back in a coffin."
Campbell's comments, widely reported on the front pages of several British newspapers on Monday, follow announcements from the families of two black England players who said they would not go to the championship.
The brother of midfielder Theo Walcott has tweeted that he and his father would not go because of "possible racist attacks" while the family of his Arsenal team mate Alex Oxlade-Chamberlain have also shelved their plans to watch England.
"I think that they were wrong," he said. "What they should say is that if you want this tournament you sort your problems out.
"Until we see a massive improvement, that you have sorted it out, you are never going to get the tournament. You do not deserve these prestigious tournaments in your country."
"I believe the tournament will be a great success, and I trust the fans will behave, but we shall see. There are problems at every tournament, always," said the Frenchman.
"We also have problems with hotel prices in Ukraine, and we have raised our serious concerns. We do not want fans to stay away for any reason, and I hope they do not."
In a statement, UEFA said: "Euro 2012 brings the spotlight on the host countries and clearly creates an opportunity to address and confront such societal issues.
"UEFA's zero-tolerance approach to racism is still valid both on and off the pitch and ultimately the referee has the power to stop or abandon a match should racist incidents occur."
