The obvious intent of this is that the sound of a woman's footsteps are non-tznius. I am no expert in tznius, so I don't know if there is any halachic discussion of this anywhere, though I would doubt it.
I see that HaEmtza is bothered by this particulat tznius suggestion as well.
In all likelihood, especially since she also promoted Rabbi Falk's book Oz veHadar Levushah, she got it from there. Indeed, he discusses it on a page not in the online preview of the book, on page 347: "Not to walk with loud or sophisticated footsteps"
I have not seen his derivation, so I am going to guess at this. I think I've developed some sense at how he goes about, going against Chazal by innovating new halachos, by darshening pesukim in Navi by himself. If you have seen the text in the book, please help me by telling me if I am right, or if I am wrong, telling me what he actually says.
There is a strong likelihood that he gets it from a pasuk in Yeshaya, since this is a text he commonly uses when inventing his tznius chumras. In Yeshaya 3:
One can focus on וּבְרַגְלֵיהֶם תְּעַכַּסְנָה. That phrase can mean different things. For example, Rashi, following the gemara, understands this as spraying venom:
The pasuk is thus: And the Lord said: Because the daughters of Zion are so haughty; and they walk with neck stretched forth, and winking eyes; walking and raising themselves they walk; and with their feet they spout "venom."
and with their feet they spout venom: When they would pass in the street near Jewish youths, they would stamp their feet and hint to them of the affection of the adulteresses, in order to arouse their temptation, like the venom of a serpent. עֶכֶס is the venom of a serpent.
This "spouting venom" involves stamping. With rubber soles you cannot stamp loudly. But these were the actions of women going out of there way to seduce, and be adulteresses. This does not mean that other, stamping is problematic; of that in a culture where men do not understand the "hint" of stamping. And if they intend nothing by the stamping and the men don't intuit anything from it, who says it is a problem? And if it is just regular walking, rather than emphatic stamping, who says it is problematic. Show me a gemara assuring stamping, or of walking with non-rubber soles.
Chazal, in Shabbat 62b, interpreted the pasuk slightly differently:
And making a tinkling [te'akasnah] with their feet: R. Isaac of the School of R. Ammi said: This teaches that they placed myrrh and balsam in their shoes and walked through the market-places of Jerusalem, and on coming near to the young men of Israel, they kicked their feet and spurted it on them, thus instilling them with passionate desire like with serpent's poison.Thus, they interpret it as kicking their feet specifically to distribute the perfume which they previously placed in their shoes. This is not the case of the conduct of women walking today without rubber soles on their shoes. (Rashi probably deviated slightly from the gemara in the interests of peshat, since spraying poison/perfume does not seem to be the most peshat-oriented explanation.)
We do have Ibn Ezra and Radak who speak of what is by their feet making noise. But we don't establish halacha based on a peshat given by a medieval commentator, just as we don't change halachic times and say night follows day, despite Rashbam's interpretation of Bereishit like that, or stop wearing tefillin if Rashbam says that on a peshat level it is allegorical.
What do Radak and Ibn Ezra say? Well, Ibn Ezra on תְּעַכַּסְנָה writes that they put achasim, perhaps anklets, or perhaps spurs, on their feet, like men who ride. (That it mentions men who ride makes me think spurs, and in medieval times they had spurs.) Then he mentions that some people say that the anklets/spurs (achasim) made noise.
But these were specific decorations, perhaps for the particular intent of making noise, and specifically with the intent to entice the men. This is not the same as a normal women walking with normal shoes, that happen to make normal noise when one walks. Ibn Ezra does not say that these are shoes without rubber soles.
Radak also understands these to make noise, but a tinkling noise. He says that they would knock their feet with the achasim which were on them, which were bells, and produce a sound.
Surely this deliberate action, with bells, is not the same as non-deliberate action, with normal shoes which normal men and women wear.