Shelter Scotland also found almost 10% of those concerned were worried about meeting payments in January.
The charity said it had been contacted by dozens of people every day who felt "overwhelmed" by mounting rent or mortgage bills.
The survey of more than 1,000 people was carried out by YouGov.
The 25-34 age group had the highest number of respondents, with 36% stating they were worried about paying rent or their mortgage.
The survey also found that almost a third of people have already cut back on winter essentials such as fuel and clothing in order to cover rent and mortgage payments, while 5% of those questioned have used savings meant for Christmas to pay living costs.
Shelter Scotland recently revealed another rise in the number of calls handled by its national helpline last year, while an average of 81 households were assessed as homeless every day.
Adam Lang, head of policy at Shelter Scotland, said: "It's a worrying sign of the times that so many people are starting the new year worried about how they'll pay for their rent or mortgage in 2016.
"Every day we hear from people who are living on a knife edge, feeling overwhelmed by mounting rent or mortgage bills, as the increasing pressure of sky-high housing costs continues to take its toll.
"We understand it is all too easy to bury your head in the sand hoping the problem will go away. But if you're in this situation, it's so important to remember you're not alone and that help is at hand.
"Advice from Shelter Scotland is only a click or a phone call away - so get advice early to prevent things from spiralling out of control."
More than a quarter of homeowners or tenants are worried they may not be able to cover the cost of their housing during 2016, according to a survey.