Mr Putin signed a new bill into law in June banning "homosexual propaganda" - making it illegal to give anyone under 18 information about homosexuality.
Anyone deemed to be promoting gay rights, or "non-traditional relationships" to young people in Russia could now be arrested and fined, or in the case of foreigners, detained and deported.
Critics say the law is intentionally vaguely-worded and part of a broader crackdown on gay rights in Russia.
The country’s sports minister has said the law will apply to athletes and spectators at the Sochi games, due to be held in the Russian Black Sea resort next February.
In his open letter, Stephen Fry said: "The IOC absolutely must take a firm stance on behalf of the shared humanity it is supposed to represent against the barbaric, fascist law that Putin has pushed through the Duma.
"An absolute ban on the Russian Winter Olympics of 2014 on Sochi is simply essential.
"Stage them elsewhere - in Utah, Lillyhammer, anywhere you like. At all costs Putin cannot be seen to have the approval of the civilised world.
Despite moral weaklings like President Obama who thinks that we should still participate in the Olympics to be held in Russia the homophobia of which matches that of many backward African countries, I think it is essential that we make a definitive moral statement that once and for all takes the moral high ground. And the only moral high ground one could take is to boycott Russia's hosting the Olympics!
Should we participate in the Olympics in Russia, we would be complicit in lending legitimacy to Russia's systematic and legally enshrined imprisonment of LGBT people and allies; giving tacit permission for its citizens to bash and even kill LGBT people.
The best way we can show our revulsion at Russia's legalized homophobia, as well as support all LGBT people and allies everywhere, is to make a definitive statement that we will have no patience with any country that would cause untold harm to its LGBT citizens. Moreover, by holding the Olympics in a place other than Russia would cost Russia a lot of credibility, and we have seen in the past that one of the best ways to make a definitive statement, and even some gains, about matters of equal rights is to publicly embarrass the oppressor.
And to put an additional and finer point on it, any LGBT athlete or ally, despite their rationalizations, who participates in, or attends, the Olympics should they be held in Russia, is the equivalent of an Uncle Tom during the eras of slavery and segregation in the U.S.