Stephen Fitzwater was sitting among the Chelsea fans at last month's FA Cup semifinal when he was overheard abusing Drogba, who is black.
Other Chelsea fans reported Fitzwater to stewards at Wembley and he was arrested by police.
The 55-year-old Fitzwater was found guilty at Hendon Magistrates' Court on April 30 and banned Wednesday.
The ban covers all football matches in Britain and overseas.
Chelsea has also banned Fitzwater for life and says "the overwhelming majority of our fans abhor all forms of discrimination."
Chelsea beat Tottenham 5-1 to reach the FA Cup final.