A third (38%) admitted to being worried about paying for their heating bills this winter, while more than three-quarters (77%) of people in the UK also believe energy prices are set unfairly.
The survey of 1,035 adults, carried out by ComRes between August 30 and September 1 for BBC Radio Five Live, also revealed strong support for renewable energy.
It found 84% were in favour of more solar panels in the area where they live, while 67% would welcome windfarms.
A third of people (33%) would support controversial method of fracking, which increases to nearly half (48%) if the community was compensated.
Earlier this summer a series of anti-fracking protests took place in Balcombe in West Sussex where exploratory drilling has begun.
While 47% support nuclear power, only 31% would want it in their area.
And 67% said they would support more coal, oil and gas stations being built in the UK if it brought energy prices down, suggesting the importance of reducing energy prices for families.
Shadow energy secretary Caroline Flint seized on figures released last month that showed only 132 Green Deals have been agreed - despite some 58,000 assessments being carried out on homes.
The Green Deal is designed to allow energy-saving improvements to be made to homes without having to pay the costs upfront. Instead, a loan is repaid through installments on electricity bills.
Energy Secretary Ed Davey said the Government would publish its fuel poverty strategy early next year, but blamed Labour's failure to invest in energy for the "massive cost pressure" on bills.
"With our emphasis on competition, on helping vulnerable consumers directly and with our energy efficiency policies, the coalition is delivering in difficult times for people, when Labour failed to deliver in easy times," he said.