He said that UEFA was wrong to choose Poland and Ukraine to host the tournament as it put supporters in danger.
The former Arsenal and Tottenham defender 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.''
His warning came as the family of two black England players said they would not go to Euro 2012.
The brother of midfielder Theo Walcott tweeted that he and his father would not be flying out to support the star because of "possible racist attacks''.
And Alex Oxlade-Chamberlain's family have shelved plans to see their 18-year-old son play in Ukraine and Poland due to safety concerns.
The BBC documentary, to be screened on Monday night, investigates violence and racism at football matches in the host countries.
It contains footage of fans giving the Nazi salute, taunting black players with monkey noises, anti-Semitic chants and a group of Asian students being attacked at the Metalist Stadium, Kharkiv, one of the venues hosting matches in Ukraine.
Campbell said UEFA, football's European governing body, should not have allowed Poland and Ukraine to host the competition, which begins on June 8.
He said: "I think that they were wrong, because 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.''
Nick Lowles, from UK-based anti-racist group Hope Not Hate, was also in Poland monitoring incidents of racism.
He said: "I think increasingly the positive thing about English football is that increasingly black and Asian fans have been travelling to support England and I am concerned that they will be targeted by racists and fascists and anti-Semites in Poland and Ukraine.''
UEFA said in a statement: "Uefa 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.''