DBS checks are a record of an individual's criminal convictions and are used by employers and organisations to assess a person's suitability for a particular job or position.
They are a requirement in many roles, particularly for people in contact with children or vulnerable people.
Offenders are legally required to disclose any previous names they have lived under, but failure to do so could result in the offences linked to a previous identity not appearing during a check.
Ms Galvin has faced accusations of transphobia online – something she rejected. She said she wanted to help the trans community too.
"This man beat my little brother black and blue. He neglected us as children, he was a violent man without the sexual abuse," she told the BBC.
She added that there were so many ways in which someone like her father "could take hold of a very vulnerable group in society and abuse it and use it for his own gain, which is what he did to me as a child."
"I'm just trying to protect people because I was never protected," she added.
Ms Galvin had suffered sexual abuse at the hands of her father for as long as she could remember.
"He groomed me," she said. "It was the whole kind of textbook 'I love you, nobody else does, we'll run away together, we'll be happy'. As a child you're fed these things from day one."
As she grew older, he threatened to turn the abuse on her younger sister if she did not co-operate.
Eventually, police uncovered what was happening when they found images of Ms Galvin that her father had been sharing with other paedophiles online.
Her father was convicted of four counts of sexual activity with a child family member, three counts of causing or inciting a child to engage in sexual activity and two counts of distributing indecent images of a child.
The Ministry of Justice (MoJ) said the risk of offenders changing their identities was mitigated by court orders, intelligence sharing between organisations such as the police and other government agencies, and that data was checked both manually and through the use of algorithms during the application process.
However, Labour MP for Telford Shaun Davies, believes the system needs to become more robust and said the government would take action.
"The whole DBS system needs to be reviewed and that is what the government is committed to doing to ensure that the system is fit for purpose," he said.
"I've also been pushing for tougher legislation around when an offender needs to update their records and I'm glad that the new policing bill that's going through Parliament will put a legal duty on offenders having to update their names to both the police and the probation service.
"Not to change their name and then tell the authorities, they'll have to do it in advance."
The MoJ added that the Criminal Justice Bill would allow the police to restrict a sex offenders' ability to make changes to their name on identity documents.
