While it's hard to speak with certainty for this time period in particular, it's unlikely Jainism was the dominant belief system in India at this time. A more convincing argument could be made for Buddhism as the dominant tradition among the ruling class, while the faith of the people had likely remained eclectic.

The period in question (~769 CE) falls within a relatively poorly documented period in North Indian history. The political turmoil following the death of Emperor Harshavardhana (c. 590–647 CE) of the Vardhana Dynasty led to a roughly 150 year gap in extensive contemporary writing. By contrast, the reign of Emperor Harshavardhana is among the most extensively documented in Indian history, with surviving records from multiple sources. The most important of these are the Harshacharita ("The Deeds of Harsha"), written by his court poet Bannabhatta, and the writings of the Chinese monk Xuanzang, who visited the court of Harsha and travelled extensively through India. A contrasting text roughly dating to after this fallow historic period is the Shankara Digvijaya, detailing the exploits of the Hindu philosopher Adishankara (born c. 788 CE). Overlaying these texts gives us some context for the religious makeup of India during the period in questions.

It's worth remembering that the faith of the lay public probably remained eclectic, drawing from disparate traditions, and moving fluidly across them. The ruling elite patronized religious institutions of all faiths, and this seemed to be reflected among the public. For example, decorative panels offered at Buddhist stupas demonstrate that Hindu merchants were also keen to seek the blessings of the Buddha. Emperor Harsha's court poet Banabhatta records Harsha as a devout Shaivite, while Xuanzang notes him to be a devout Buddhist. Both sources record Harsha donating generously to holy men regardless of their tradition. Xuanzang records a yearly charitable event for monks. While four days each were set aside for Hindu and Jain monks, 16 days were set aside for the Buddhist monks. It's unclear if the disparity was due to imperial favouratism, or simply reflects a skew in the number of holy men from each tradition. If Banabhatta is to be believed that Harsha was a devoted Hindu, then Xuanzang's record of the number of days would suggest that the Buddhist monks were much more numerous in the empire. This version of events is also supported by the concerns of writings of the traditions themselves. For example, the Hindu texts composed leading up to this period (The age of the Sutras) are devoted to apologia for their own philosophical school, and polemics against other philosophical schools, including other Hindu schools and the Buddhists (and to a lesser extent, the Jains). Contrast this with the writings of the Buddhists during the same period, and they are relatively unconcerned with the Hindus, who are treated as a philosophically spent force. The force of the arguments are reserved against rival Buddhist schools. Whether this is the blossoming of Buddhist commentary writing in general, or reflects the headspace of Buddhist writers as chiefly occupied with the heresies of other nominal Buddhists is anybody's guess.

If we now incorporate the religious landscape as laid out in the late 8th century Shankara Digvijaya, the Buddhists are established as the dominant religious tradition in the country. Shankara's writings are aimed towards solving the theological arguments raised by the Buddhists, and they are his chief rivals. It's also likely that the cultural importance of the Buddhists as an opposing force is overblown in the Digvijaya, to further enhance the cache associated with Shankara for defeating them. Nevertheless, when the Digvijaya was composed in the generation following Shankara, it wouldn't make sense to prop up the Buddhists as the chief rivals if they hadn't been a major force within living memory. This disparate sources of evidence would suggest that the Buddhists had been a major religious tradition at least upto the 8th Century CE. This time period also represents a major shift in the Buddhist schools in Northern India at the time, with the Theravada school losing out to Mahayana, and eventually Vajrayana.

None of these sources hold the Jains as a significant political or cultural power during this time period. While they were significant communities of Jains in South India, I am unfamiliar with a widespread Hindu or Buddhist engagement with them, though I could be mistaken.