Probably not.

The long-standing perception that the plague affected the east as much as the west seems to be a legacy of the 'third plague pandemic' of the 19th century, which was concentrated in China and India. For example, in 1881 a professor of medicine, August Hirsh, wrote that;

Until quite recently, everyone considered it settled that Persia was the eastern limit of the area of plague on Asiatic soil, beyond which, during the last five centuries at least, it had never penetrated. But the latest observations in India and China have completely upset that opinion.

There is, however, no unambiguous description of the bubonic plague in either Chinese or Indian records before the early 17th century, after which vivid descriptions appear almost simltaneously in both regions. While deadly epidemics are recorded for 14th century China, there is no solid evidence that these were instances of plague, and much reason to believe that they were the result of the many devastating wars, floods and displacements that were afflicting China at the time. In India, on the other hand, there are no records at all of major outbreaks in the mid-14th century, despite the survival of detailed Delhi Sultanate chonicles for this period. In fact, these chronicles record general prosperity and a growing population. While European records tell of abandoned fields and deserted villages, Indian chroniclers could write that;

There was not a single ruined village in the Doab region . . . and no land was without cultivation. Fifty two new settlements sprang up in the Doab region during [Firuz Shah's] days. In regions other than Doab, the situation was the same.

Even taking into account the propagandistic nature of these chronicles, they make it hard to imagine anything on the scale of the European/Middle Eastern plague, especially when contrasted with the abundant descriptions of plague in contemporary Middle Eastern and later Mughal records, both products of essentially the same Islamic literary culture as the Delhi chronicles.

Why the plague didn't spread from the Middle East to India, despite very strong connections between the two regions, is difficult to explain, especially given the eventual spread of the plague in India from the 17th century on. Sussman suggests that the most efficient plague vector, the oriental rat flea (Xenopsylla cheopis), may have spread from Egypt to India in the 17th century, though others think it spread only in the 19th century. It's apparent, moreover, that the plagues of the 14th century were different in nature from those that hit 17th and 19th century India and China, with the latter characterized by stange rat behaviour, high rat mortality, a slow pace of movement from region to region, and a much lower rate of human mortality (less than 5%). In contrast, the 14th century plague moved very quickly between regions, killed off about half the population, and was not linked to rats by contemporary records. This difference seems to signify that the 14th and 17th/19th century plagues spread through different means, perhaps due to the spread of different flea species.

Source: George D. Sussman (2011), 'Was the Black Death in India and China', Bulletin of the History of Medicine, Vol. 85, No. 3

There are no sources to clearly say it did. There is no record of significant epidemics either. We have the chronicles of Zia ud-din Barani, "Tarikh-i-Firuz Shahi" (Firuz Shah's History) from 1357. Barani lived in the Delhi Sultanate at the time when Black Death was supposed to happen but notes only one epidemic which affected the Sultan's army, during a campaign in central India in 1334. He uses the Arabic word "waba" to describe the disease and that's usually translated as "cholera" although it's a general term for an epidemic. The Arabic word for plague is actually "taun". No specific symptoms were described either, so there's no way to claim this record as evidence for plague.

The period after Barani's chronicle is recorded by Sham Siraj Afif. According to him, up to 1388 there was prosperity and population growth. Only in 1619 we get a definitive record of the plague in India. It's a record by the Mughal emperor Jahangir himself, "Tuzuk-I-Jahangiri" (Memoirs of Jahangir). The description of the symptoms, the swelling of lymph glands and the other detail make it clear that it is indeed the same disease. The conditions of the season itself also correspond with today's understanding of how the plague develops. And on top of that, Jahangir notes how there are no records or memories of the same disease being noticed in the country in the past either.

There are several possible explanations of why the plague did not spread to India in the 14th century but they all have flaws and don't fully cover all possibilities. Most probably it would have been a combination of factors rather than one single thing. It could have been the distance, or the mountains on the way, or the climate. But unless somebody tells us for sure in a proper contemporary record, there's no way to know.