She makes this claim at several occasions in her book The Hindus: An Alternative History: that Sanskrit, as we understand it, was the language of the priestly caste (the Brahmins), invented for the purposes of religious rituals and operations, and is therefore younger than the "natural" language which she calls "Prakrit". On page 5, for instance she says:
It must have been the case that the natural language, Prakrit, and the vernaculars came first, while Sanskrit, the refined, secondary revision, the artificial language, came later. But South Asianists often seem to assume that it is the other way around, that the dialects are “derived from Sanskrit,” because Sanskrit won the race to the archives and was the first to be written down and preserved, and we only encounter vernaculars much later. So we say that Sanskrit is older, and the vernaculars younger. But Sanskrit, the language of power, emerged in India from a minority, and at first its power came precisely from its nonintelligibility and unavailability, which made it the power of an elite group. Walt Kelly’s Pogo used to use the word “Sam-skrimps” to describe highfalutin double-talk or manipulative twaddle. Many Euro-Americans mispronounce it “Sanscript,” implying that it is a language without (sans) an (intelligible) script, or “Sand-script,” with overtones of ruined cities in the desert or a lost language written in sand.
The idea that "Prakrit" (which is a term primarily used to refer discretely to the different Middle Indo-Aryan languages spoken from the 600s BCE to about 1000 CE) is older than Sanskrit is rather problematic, because it has (to the best of my knowledge) been definitively established that the Middle Indo-Aryan languages were descendants of the Old Indo-Aryan languages (of which Vedic Sanskrit is the only attested form). She seems to completely discount Vedic Sanskrit (which is mentioned only twice in the entire book) and of its being (again, to the best of my knowledge) the probable provenance of the Prakrits. She also draws no distinction between Vedic and Classical Sanskrit, or of the stages of development between the two forms. This leaves the reader with the impression that Classical Sanskrit (which she refers to simply as "Sanskrit") was a limited and artificial tongue derived from the Prakrits, instead of both being mutations of Vedic Sanskrit. (She also provides no linguistic evidence to support her suppositions.)
Is that impression correct? Am I missing something crucial?
Doniger seems to mean that Sanskrit was a "constructed" (or "refined") language while Prakrit was not. Of course, Doniger should have been more specific in what she means by "Sanskrit," since that term has been applied carelessly to many different forms of speech. By "Sanskrit" Doniger seems to mean Classical Sanskrit, which is a standardized register of late Old Indo-Aryan that came into being during the late 1st millennium BCE, often associated with the grammarian Panini. By that time, all late Old Indo-Aryan dialects were incomprehensible to Indians, as Indo-Aryan languages had already evolved into the dialects we call "Prakrits." The various Prakrits are indeed older than Classical Sanskrit, and are attested in the epigraphic record much earlier. The oldest inscriptions in India, produced under Ashoka in the 3rd century BCE, were composed in Prakrit and not in Sanskrit.
I think the reason you are confused is because you think Doniger is saying that Classical Sanskrit was linguistically derived from Prakrit. If that's what Doniger is saying, then she is wrong, but that's not the impression that I got from reading the excerpt you quoted. Rather, she seems to be saying that Classical Sanskrit came into being through an "artificial" resurrection and standardization of late Old Indo-Aryan, and not through a "natural" or organic evolution from Prakrit. I think that's what she means when she says that Sanskrit's power "came precisely from its nonintelligibility and unavailability."
A disclaimer, I am familliar with the book, but have not read all of it nor am I remotely fluent in Sanskrit, Pali, or any Prakrit. I am not a philologist l, nor am I well versed in the subject. That said, my background is in Indian religion and classical India.
Doniger is suggesting that Sanskrit, as in Old Sanskrit, is younger than Prakrit and was developed as a deliberate means of keeping the Vedas and other sacred texts from most, since they could not understand the language. Her assertion is that Sanskrit is a constructed literary language which was never widely spoken. It is unclear (to me) as to whether by Sanskrit she is referring the Rig Vedic language or Sanskrit as in the Classical Sanskrit codified by Panini and expanded by later people. If the former, then she is making a strong statement, there is a general concensus of tha it is in fact older due to being archaic and being similar to the Avestan language. If the later, it is a reasonable, if somewhat controversial claim (to Hindu fundamentalists).
I would also reccomend reading literature other than this as well. Doniger is a somewhat divisive figure within Indian scholarship and this book presents a quite different view from the standard one at times. I've also read criticisms from historians of her misrepresentation of British colonial figures, ritual theory and practice, and Mughal influence on popular Hinduism in academic journals. She also tends to loose focus: why so much time on the Mughal rulers? That said, her work is widely praised for its depth and breadth, and with a volume so large minor errors or innaccuracies are expected. What I've read from it (mostly the early parts) is well written and engaging. It isn't the definitive work, but hardly any is, and she succeeds in presenting an alternative view.