According to my Oxford Classical dictionary, the toga was originally also worn by women without any stigma of prostitution attached. That being said, later down the line it definitely had this prostitution label attached.

A great number of things would have had to change in Roman and vestigial Etruscan society for women of good standing to no longer wear the toga. And one of those changes would have come in the form of drama. And I mean that in the most literal sense of "the stage." The togata was a type of comedy written at Rome in the 2nd century BC (also known as tabernaria) in which characters -- though not all characters -- would have worn the toga. It may have been here that prostitutes were portrayed as wearing the toga as a source of comedy, and which became popular enough a motif.

That being said, we know that by the 1st century BC with the writings of Cicero and Sallust that the toga worn by women was already a symbol of prostitution (for further reading, see Sallust's portrayal of Sempronia, BC 25, and Cicero's portrayal of Clodia in his Caeliana).

Alongside togata there is also evidence of the toga being worn by women as a sign not only of prostitution but of adultery and masculinity. Antique scholastic notes about Horace say the following:

"Matronae who were divorced by their husbands because of infidelity used to take up the toga, having put aside the stola because of their disgrace. Indeed, the toga is appropriate for the prostitute. This is because they used to solicit customers like this, dressed only in dark-coloured togas, in order to be distinguished from respectable women, and for this reason those women who had been convicted of adultery used to wear this garment. In other words, women who had been convicted of the crime of adultery were said to walk about in public wearing togas. Others say that the women wearing the toga is a freedwoman, because previously freedwomen wore togas, but matrons the stola." (ad Hor. serm. 1.2.62-63).

Thomas McGinn notes however that there is no other text about freedwomen wearing the toga and that this particular section of the text is likely a red herring.

This being said, not all prostitutes would have exclusively worn the toga. Prostitutes would have had a wide range of clothing that they wore. Still we can deduce that the wearing of the toga by women became a symbol of public shame that focused on the distinctly sexual: women who sold their bodies for sex, or who were accused of adultery.

Sources: Oxford Classical Dictionary "Prostitution, Sexuality, and the Law in Ancient Rome" by Thomas McGinn, Oxford University Press, 2003.