Proposals that would make it an offence to pay for sex are to be considered by the Scottish Parliament's Justice Committee.
The bill would criminalize those who use prostitutes.
A similar proposal in 2010 was opposed by ministers, who feared it would push the sex trade underground.
Critics of such legislation believe that making workers in the sex trade less visible to the authorities would place them in greater danger.
But Ms Grant said that it could tackle the serious criminal activity associated with prostitution.
She said: "What it does is recognized that women who are involved in prostitution are indeed victims.
"It also puts the onus on the people that purchase that service to realize that they are dealing with people who are vulnerable and are victims, and they need to take responsibility for their own actions."