The new figures from the HFEA showed that 17 men had fathered at least 30 babies each between 1991 and 2015. A further 104 men have fathered between 20 and 29 babies, and 1557 between 10 and 19. More than 6000 have created nine or fewer.
Although donated sperm is screened for major diseases such as HIV, hepatitis B and C, Creutzfeldt-Jakob disease (CJD), Huntington's and cystic fibrosis, clinics do not yet test for genes which increase the risk of certain cancers or conditions such as Alzheimer's disease.
Up to 60 per cent of women who carry a BRCA1 mutation will develop ovarian cancer in their lifetime, compared to just 2 per cent of the general population, and up to 90 per cent will develop breast cancer.
The charity Ovarian Cancer Action warned that the current genetic testing process is only carried out if there is family history of cancer and donor-conceived children can face 'enormous barriers' in accessing checks.
Marie-Claire Platt, Head of Campaigns at Ovarian Cancer Action, said: "Without proper screening for hereditary cancer gene mutations, donors could unknowingly pass on an inherited risk of ovarian, breast, bowel and other cancers.
"This is particularly worrying for couples who are trying for a baby who might not be aware of the implications for their sons or daughters in the future.
"We urge the regulators to take action and include hereditary cancer gene mutation screening in the process of sperm and egg donation."
Under HFEA rules donors are not allowed to create more than 10 families, which means many of the youngsters will live with their half-siblings, meaning all may end up carrying a defective gene.
In 2013, it was discovered that Danish sperm donor Henrik Koch may have unknowingly passed on a cancer-causing illness to 50 per cent of the children created using his sperm.
This does bring up an interesting issue about how much we should be screening donors
Both men and women can pass the BRCA1/2 gene mutations and there is a 50 per cent chance of a child acquiring it if one parent is a carrier. However if both are carriers the child will definitely inherit.
In the US, IVF clinics are increasingly using enhanced screening techniques to make sure faulty genes are not passed on.
Geoff Trew, Clinical director of the Fertility Partnership, one of Britain's largest IVF providers, and researcher at Imperial College, said he believed enhanced screening would soon be brought in for sperm donors in the UK.
"This does bring up an interesting issue about how much we should be screening donors. Perhaps we should be doing a bit more." he said.
"I think over the next year we will start to see clinics brining in enhanced screening. It could highlight if there was a very big risk of passing on a defective gene, such as it both parents had it. The interaction between the genes is important."
Since 2005, donor conceived children have been able to find out the identity of sperm donors but those born before cannot find out about their biological fathers.
However Allan Pacey, Professor of Andrology at the University of Sheffield, said bringing in further screening for donors could leave Britain facing a shortage. Around one third of donated sperm for IVF is currently imported from abroad.
"There isn't such a thing as a perfect donor," he said.
"Everyone has a defective something, and if we are going to start screening everything out you may get to a point where we have no sperm donors left at all."
