Was Buddhism the dominant religion of India for 1 thousand years?

It is a broad statement, and difficult to prove in terms of population percentage practicing Buddhism in the whole subcontinent as opposed to being patronized by monarchs. In fact Amartya Sen makes it amply clear in his book that he refers to the fact that everyone, including Chinese travelers, referred to the subcontinent as a "Buddhist Kingdom". However, Buddhism did flourish from 6th century BC. Here is one point of view that is in line with Amartya Sen's:

In regard to India’s past, we will argue that the decisive period for the formation of the continuous ‘thread’ of history was the first millennium BCE, and that to a very large degree the thousand years after this represent a civilisation dominated by Buddhism: ancient India was not ‘Hindu India’ but ‘Buddhist India’.
Source: Buddhism in India by Gail Omvedt

Keep in mind though that there was no such thing as "Hindu" in those days. This term is an Arabic/Persian term used to describe the people east of the Indus. The Brahmanic traditions came into formalised existence as an umbrella religion (also as per Amartya Sen's book) much later, although diverse groups existed.

The same can be said about early Buddhism.

...‘Buddhism’ is not a religion in the conventional sense, ‘Brahmanism’ also was more than just a religion. It included a required social practice (varnashrama dharma) and it absorbed, or rather co-opted and reinterpreted, many indigenous religions and cults.
Source: Taranatha's History Of Buddhism In India

Another evidence to suggest a nebulous cult rather than a "religion":

During the first and second centuries after the Nirvana, Buddhism could hardly be distinguished from other ascetic movements. It was evidently in the Maurya period that Buddhism emerged as a distinct religion with great poten­tialities for expansion. But even at the beginning of this period, its activities were mainly confined to Magadha and Kosala. Small communities of brethren may have come into existence also in the West, in Mathura and Ujjayini. At the time of the Second Council, which was held at Vaisali about a hundred years after the Buddha, invitations were sent to communities in distant places like Patheya, Avantl, Kausambi, Sankasya and Kanauj. Mathura had become an important centre of Buddhism in the early years of Maurya supremacy.
Source: 2500 Years Of Buddhism

However, as with most religions, power and hegemony grow with time, and the need for constructing authority, identity and community has to be provided by narrative and legend.

As Buddhism outgrows the narrow circle of the early Magadhan communities and expands to the north-west, the need is felt for new legends to justify the authority of the new communities and the mahatmya of their new centres. The most prominent of these new centres are Mathura and Kashmir. Mathura is the centre of the Sarvastivadins and Kashmir that of those who call themselves the Mula-sarvastiva-dins. The typical literary product of the monks of Mathura is the Asokavadaila and the most archaic form of this, according to Przyluski, is the Asoka-rlija-sutl'a, now preserved in Chinese translation as A-yu-wang-king. The typical literary product of the monks of the north-west (Kashmir) is the Vinaya of the Mula-sarvastivadins.

In support of "Buddhism" as a dominant religion, we can undoubtedly say that art and architecture discovered from the past indicate overwhelming Buddhist influence for over a millennium. The earliest religious architecture are Buddhist— viharas, stupas, cave paintings, chaitya halls, monasteries, and also statues. There is no Hindu temple until the time of the Guptas, and even these were small.

While Brahmanic religious literature exists from this period, such as the Upanishads, the Dharmasashtras, the Mahabharata and the Ramayana(up to 1st Century AD), Pali Buddhist literature from this period, now surviving mostly outside India, is much greater in volume. Also, much of the earliest Sanskrit literature was Buddhist, such as that of Asvaghosha and early Mahayana. Other literature is either secular or Buddhist-influenced, such as Tamil Sangam literature and Kavya literature, covering the period up to the 6th century AD.

With special reference to Tamil Nadu in the South:

From the 4th to the end of the 6th century, the rulers were supporters of non-Brahmanic religions: it was the period of great literature, the kavyas and the great didactic poem Kural, all influenced by or openly propagating Buddhism and Jainism. The ‘Hinduisation’ of Tamil Nadu took place only after the Brahmanic revival under the Pallavas in the 7th century. This period saw the rise of militant bhakti movements focused on Shiva and Vishnu strong anti-Buddhist and anti-Jain propaganda, as well as the sophisticated campaigns of the Vedantic philosopher Shankacharya in the 8th century CE.

Chinese travelers’ accounts show, with Harsha in the early 7th century, a king could again be characterized as Buddhist, though he issued coins depicting Shiva as well as the Buddha. Many such factors indicate that the early, classical age of India was very much Buddhist dominated.

According to the Jatakas, Indian merchants went to Babylon, which was known as Baveru, to southeast Asia, and to Sri Lanka. Traders along the Silk Road were Buddhist too, carrying the religion well into China, Middle East and Central Asia.