Emma Thompson, Stephen Fry and Aisling Bea are among celebrities calling on the five of the UK's biggest High Street banks to stop financing new oil, gas and coal projects.
It follows criticism that HSBC, Barclays, Santander, NatWest and Lloyds are funding "fossil fuel expansion" despite making green pledges.
Businesses and charities like Greenpeace also back the campaign.
HSBC and Barclays said they were helping their clients to cut emissions.
The Make My Money Matter campaign points to research by environmental charity Rainforest Action Network, which claims that between 2016 and 2021, HSBC, Barclays, Santander, NatWest and Lloyds funnelled almost $368bn (£298bn) towards the fossil fuel industry.
It added that in the same time period, the lenders financed the 50 companies making the biggest investments in oil and gas projects to the tune of $141bn.
It added that while HSBC and LLoyds had made "welcome new announcements" on stopping direct finance for new fossil fuel expansion since then, "there is a long way to go".
"HSBC was this month found to have provided $340m to a company opening a new coal mine in Germany," it said.
The campaign, which is also backed by actor Mark Rylance and musician Brian Eno, urges the public to sign an open letter asking the banks to stop directly financing projects that expand fossil fuel use, or end relationships with clients that do.
The campaign's founder, filmmaker Richard Curtis, said he wanted to put "a fire under the banks".
"It's clear that new oil and gas fields are not only hugely damaging to the planet, but they're also wildly unpopular with the public," he added.
Almost one third of HSBC, Barclays, Santander, NatWest and Lloyds' customers surveyed by the campaign said that they would switch bank if they discovered that theirs was financing the expansion of fossil fuel projects.
Over 85% of customers at the five banks surveyed said they did not think that their bank was doing enough to tackle the climate crisis.
TV presenter Chris Packham CBE said that financial institutions had an "enormous ethical and moral responsibility" to start withdrawing funding from organisations that damaged the climate and biodiversity.
High Street banks have made steps to loosen their ties to environmentally damaging companies.
HSBC told the BBC it no longer provided new finance or advice for new oil and gas fields. It added that "supporting clients in high-emitting sectors to decarbonise will have the most significant impact on emissions reduction".
Barclays said it would reduce its support for "carbon-intensive" companies over time if they were "unable or unwilling to reduce or eliminate their emissions". Since January 2019, it has not provided loans for the development or expansion of coal-fired power stations or greenfield thermal coal mines.
NatWest Group said the bank no longer lent to or underwrote coal or major oil and gas producers "unless they have a credible transition plan in line with the Paris Agreement". It said this had resulted in a £1bn drop in fossil fuel financing at the bank.
Santander said it is "fully committed" to supporting the transition to net zero and won't finance new projects to produce oil.
Britain's biggest domestic bank, Lloyds, announced it would stop "direct" financing to develop new oil and gas fields in October. The BBC has also contacted Lloyds for comment.
Under the 2015 Paris Agreement, 197 countries agreed to try to keep temperature rises "well below" 1.5C to avoid the worst impacts of climate change.
Experts say that to achieve this, countries must have net zero emissions by 2050.