"Stay at home, watch it on TV. Don't even risk it ... because you could end up coming back in a coffin," Campbell, who is black, told the BBC.
The families of two of England's black players, Theo Walcott and Alex Oxlade-Chamberlain, have already indicated they are unlikely to travel to the Euros because of fears of racist abuse.
After witnessing footage in BBC's Panorama program, to be broadcast later Monday, of fans from the host nations giving the Nazi salute and taunting black players with monkey noises, Campbell said UEFA was wrong to award the tournament to Poland and Ukraine.
"I think that they were wrong, because what they should say is, 'If you want this tournament, you sort your problems out. Until we see a massive improvement ... you do not deserve these prestigious tournaments in your country.'"
The documentary also shows a group of Asian students being attacked at the Metalist Stadium in Kharkiv, one of the host venues.
In a statement, UEFA — European football's governing body — said the tournament "clearly creates an opportunity to address and confront such societal issues."