The poorest paid are particularly worried, the research found, but also the least likely to speak up about it.
Younger workers are also less likely to raise a complaint, the Resolution Foundation said.
The widespread concerns come despite government advice on making workplaces Covid-secure, researchers said.
Lindsay Judge, research director at the Resolution Foundation, said: “More than one-in-three workers are worried about catching coronavirus on the job, despite the extensive steps employers have taken to make workplaces Covid-secure.
“Given many workers’ limited ability to get employers to address Covid concerns, the UK needs a strong enforcement regime to ensure that workplaces are as safe as can be.
“But instead health and safety resources have been cut, inspections have been slow, and Covid-related enforcement notices are few and far between.”
The researchers said they were also concerned about the reduction in funding at the Health and Safety Executive.
The HSE’s funding for each site it has the right to inspect has shrunk from £224 a decade ago to £100 for the current financial year, according to the report.
“The Foundation says policy makers should overturn the current view that health and safety is a ‘brake on business’ and take a more proactive approach to enforcement in the face of the pandemic,” it said.
A government spokesman said £14m of funding was given to the HSE to combat coronavirus earlier this year.
An HSE spokesperson said: “We thank the Resolution Foundation for its report, and with our partners across government we will examine its findings. We welcome the acknowledgement of our increased activity and share the commitment to ensure all employees have a voice.
“Making sure Great Britain’s workplaces are Covid-Secure is our priority; this effort will not be affected by recent additional restrictions announced across England, Scotland and Wales. We will work with stakeholders to deliver workplace health and safety during this coronavirus pandemic.
“Inspection and putting duty holders on the spot is just one part of a wide ranging regulatory approach. We use a number of different ways to gather intelligence and reach out to businesses with a combination of site visits, phone calls and through collection of supporting visual evidence such as photos and video footage.”
While a new lockdown in England is planned from Thursday, up to half of workers could still be going to work in jobs such as essential retail, education and health, Resolution said.
The research used an online YouGov survey of 6,061 adults across the UK.