Mental health problems affected everyday activities such as budgeting and paying bills, the Money and Mental Health Policy Institute (MMHPI) said.
Setting spending limits on cards and allowing people to set how banks contacted them would help, it said.
The trade body for High Street banks has vowed to improve inclusion.
In its mission statement covering 2016-18, the British Bankers' Association (BBA) said that the "crucial part of the industry commitment to raising standards" should include working with mental health initiatives.
The MMHPI said one in four people could suffer from mental health issues in any one year. It has published research suggesting that periods of poor mental health put people at risk of financial trouble.
For example it found that people with depression or post-traumatic stress disorder were likely to struggle with short-term memory, making Pin numbers harder to remember.
Those experiencing bipolar disorder or ADHD often struggled to resist impulses, potentially leading to dramatic spending sprees, it said.
People with borderline personality disorder or psychosis could find it very difficult to compare financial options and found it more difficult to plan ahead.
Extreme anxiety could also stop people opening letters or taking calls from banks.
Dan, aged in his 40s, says his finances are affected by his bipolar condition and anxiety. This led to a phobia of opening letters and answering the telephone, fearing his "safe space was being invaded".
As a result, bills went unpaid and bank charges built up, leading to debts which he is still repaying after 15 years. "I'm still playing catch-up," he says.
Although conditions are very individual, he says he would have been helped by correspondence through e-mail which would have allowed him to compartmentalise his financial matters and "take away the emotion" by dealing with it in work mode.
"Senior managers want to put measures in place, but this does not always filter down to branches and call centres," he says.
He would like to see better training in place. Clearer literature from banks, on what assistance is available for customers with mental health problems, would also allow people to make an informed decision on revealing their condition to bank staff.
The MMHPI is challenging banks to adapt some systems already available to help those with mental health problems. For example
Polly Mackenzie, the Institute's director, said: "For too long, it has been assumed that when people with mental health problems get behind on bills, or struggle to stick to their budget, it is because they are lazy or incompetent. Our research proves beyond doubt that's just not true.
"Mental health problems can severely affect consumers' ability to stay on top of their finances, shop around, or manage a budget.
"It is time for the financial services industry to adapt its services to help support people when they are unwell - just as they do to help people with physical disabilities who struggle to access a branch or engage on the phone."
A spokeswoman for the BBA said: "The industry is committed to supporting customers in vulnerable circumstances, and will consider these recommendations carefully alongside the pioneering work already done with the Royal College of Psychiatrists and others on debt and mental health issues.
"It is vital that vulnerable customers are offered the right support, although we recognise that balancing this support with respect for customers' privacy is a difficult and sensitive issue.
"The industry is continually looking at ways to help vulnerable customers, and develop alternatives to make everyday banking arrangements easier, with useful information and suggestions on different types of payments a key part of these developments."
Banks must offer basic account options, as given to other vulnerable people, to assist those struggling with mental health conditions, a think tank says.