The Carry On and EastEnders actress was diagnosed with Alzheimer's disease in 2014.
The 82-year-old said: "My heart goes out to the many, many people who are really struggling to get the care they so desperately need.
"Please join us, and sign our letter - let's do everything we can to sort this out."
The actress and her husband, Scott Mitchell, have called on people to sign the open letter to urge Mr Johnson to take action.
Dame Barbara, best known as pub landlady Peggy Mitchell in EastEnders, plans to hand in the Alzheimer's Society letter with her husband on September 2.
They are among more than 50,000 people have now signed the letter.
Mr Mitchell, who has been meeting those affected by dementia, called on politicians to fix the "social care crisis", calling it a "disgrace".
"It really seems this is the disease of punishment. Not only do people have to live with it and go through the terror and horror - and their loved ones caring for them do too, but on top of that they'll say ' Let's make it a bit more difficult'," he said in a new video.
"I genuinely feel for the people that have to give up their jobs and sell their homes, selling their parents' homes that they have worked all their lives for.
"And you do wonder, how did we get here? How did we get to a situation where there is so much disregard for people?
"The conversation is getting much louder, and much broader, and they can't ignore it. They can't ignore what's going on. It's a disgrace.
"We are calling on the public to help us end this injustice and sign our open letter that myself and Barbara will be delivering to the Prime Minister next week. This cannot go on any longer."
Talking on BBC Breakfast, Mr Mitchell said: "Every time Barbara asks me the same thing I try as much as I can to smile. Because what you've got to understand is, in her mind, that's the first time she's asked me, it's possibly about the twelfth time in the space of an hour or so and I give her the same answer, slight variations each time.
"It's very frustrating, it's hard to deal with, but at the same time, they can't help it, this is a very cruel disease for people to go through.
"And we should not be seeing people, in a way, penalised, for having this - any other illness you would have there is more care in place, there is more support but for some reason dementia seems to be the poor relative".
Alzheimer's Society chief executive Jeremy Hughes said: "Barbara and Scott's commitment to helping us achieve our goals is so inspiring.
"Not only are they going through their own deeply personal experience of dementia, they are also leading our campaign, and Scott is visiting others affected to get a broader insight from those desperately battling the care system to try and get the support and care they need.
"One million people in the UK will have dementia by 2021 - yet the system is in disarray, completely unprepared to support the growing numbers of people receiving a dementia diagnosis."
People with dementia and their families often have to spend typical care costs of £100,000 "just because they happened to have developed dementia and not some other disease", the charity says.
Mr Mitchell and Dame Barbara are ambassadors for the Alzheimer's Society.
The open letter is at alzheimers.org.uk/fixdementiacareletter
Ian Hudspeth, chairman of the Local Government Association's Community Wellbeing Board, said: "Councils are committed to improving the lives and wellbeing of people with dementia and their families. However, vital adult social care services face a £1 billion funding shortfall next year, rising to £3.6 billion funding gap by 2025.
"If councils are to continue helping people who rely on vital care and support to lead the lives they want to lead, we urge the Government to address the growing funding gap in the forthcoming spending round and to propose a long-term sustainable solution as soon as possible."
A Department of Health spokesman said: "Dementia is a major health and care challenge of our time and we're working to improve care and support to make this the best country in the world to live with the condition.
"We have given local authorities access to nearly £4 billion more dedicated funding for adult social care this year, and a further £410 million is available for adults and children's services."The Prime Minister is committed to fixing the social care system and we will outline proposals in due course."