Jonathan Jones has some simple words of advice for national artistic institutions currently feeling the financial squeeze: "If they can get money from Satan himself, they should take it." The phrase is deliberately provocative, but succeeds in reaching the heart of the debate over BP's sponsorship of the arts. The argument is straightforward enough – it's time to batten down the hatches and ignore the storm of protest, because without organisations such as BP the arts might simply cease to exist.

Responding to Jones yesterday, the artist John Jordan suggested one problem with this approach: that art risks selling its soul. BP's money is tainted, and it is hard to see how the company's reputation won't have a long-term impact on those who accept it. The spill in the Gulf of Mexico is the environmental scandal of the decade, but it won't be the last. And as BP strives to extract the last drops of oil from ever more remote regions of the planet, a whole new kind of reputational risk begins to emerge. Shocking images of oil-soaked pelicans will not be around for ever, but the consequences of climate change will be with us for the rest of the century.

Tate director Nicholas Serota needs to consider this risk carefully. Does his institution want to be associated with one of the world's biggest single sources of pollution? One that has actively lobbied to undermine clean energy, pouring huge sums into industry groups that campaign to lower carbon taxes and weaken climate legislation? BP's alternative energy business is a plaything of former boss Lord Browne that has been consigned to the corporate rubbish tip. For these reasons and others, BP is certain to remain the focus of environmental resistance and public anger for years to come. Similarly, those who choose to lend the company an air of acceptability by receiving corporate sponsorship will continue to be seen as legitimate targets for protest around the world. This movement is still in its infancy, but will only gather in strength.

The second problem simply concerns credibility. The Tate website proudly proclaims its ethical policy, announcing that it will not accept funds from a donor who has "acted, or is believed to have acted, illegally in the acquisition of funds". As lawmakers on Capitol Hill put the final touches to a series of massive lawsuits, and criminal prosecutions loom on the horizon, it is hard to find a single individual who claims that BP has acted in compliance with the law. Far more compelling, though, is the Tate's stated ambition to demonstrate "leadership in response to climate change". If ever there were a moment to show such leadership, this is surely it. Tate has a unique opportunity to demonstrate that one of the UK's most progressive institutions is prepared to take meaningful steps to show its opposition to carbon-intensive industry. Currently, it refuses to even acknowledge BP's record as an issue, relying instead on bland statements that mention only the longevity of BP's financial support. There is clearly a disconnect, and behind closed doors there must be real uneasiness in the boardroom – not to mention the membership.

The issue here is not sponsorship per se, but choices. Over the past few days a number of commentators have pointed out that tobacco companies are now seen as an unacceptable partner for any self-respecting artistic body, but for some reason oil companies are still welcome to the private view. This comes despite human rights abuses, refinery explosions, the destruction of entire ecosystems, and political interference on a historic scale. You have to wonder why. Sure, BP probably offers slightly more money than the other companies vying for the sponsorship deal. They probably don't interfere too much, either (some might say that they know a thing or two about secrecy and discretion). But the fact is that there must be a host of other companies out there who actually fit the existing ethical policy of these organisations, and a relatively small financial hit is surely worth the reputational protection such a deal would provide.

By now you might be asking what all the fuss is about. After all, it's only a small logo on a programme, a discreet thank you at the bottom of the catalogue. Jones says: "I must have seen the BP logo a thousand times on press releases and it never lodged in my mind." But ask any branding expert: it's exactly this kind of subliminal association that gives a brand its identity. Until the Gulf of Mexico disaster, BP's green sunflower was found only in carefully selected locations designed to give the company an air of clean, British authority: Covent Garden, the National Portrait awards, a new exhibition at the Tate. These are some of our best loved pastimes, and for BP this feelgood factor is simply priceless. Their executives do not sponsor the arts as a way of "giving something back", or because they truly believe in opera, or painting, or culture. They simply believe in winning political and cultural aquiescence in the ugly business of oil extraction, and the sponsorship deals allow them to do just that. The millions BP spends on our artistic institutions represents an absolute bargain. Unfortunately, it is the rest of society that is being ripped off.