The NatCen British Social Attitudes Report found 48% of the 2,878 people it surveyed were in favour of capital punishment.
It is the lowest figure since the survey began in 1983, when around 75% of people were in favour.
The death penalty was legally abolished under the Human Rights Act in 1998.
Capital punishment has long been an issue over which Parliament and the public were at odds.
In 1965, a year after the last executions in the UK, MPs voted 200 to 98 to suspend the death penalty for murder, even though opinion polls suggested the vast majority of electors wanted it kept on the statute books.
It was last debated in Parliament in 1998 during the passage of the Human Rights Act. On that occasion, a provision of the Act outlawing capital punishment for murder except “in times of war or imminent threat of war” was backed by 294 votes to 136.
In the same year, the Criminal Justice Bill removed the death penalty from the sentencing options for high treason and piracy with violence, the last two crimes remaining on the statute books that were punishable by death.
The British Social Attitudes survey has been carried out every year since 1983, with questions repeated periodically to assess how opinions change over time.
For this year’s report, NatCen interviewed a representative, random sample of 2,878 adults in the UK between August and November 2014.