You may want to direct this question to /r/AcademicBiblical, as they have a number of scholars and enthusiasts who can better field this type of question. As far as direct influences on the earliest teachings, the main move since the 70s has been towards a primarily Jewish influence, which seems obvious, but late 19th century scholarship introduced the idea that early Christianity was heavily shaped and influenced by Hellenistic and Oriental religious thought and teaching. Modern scholars like NT Wright, Mark Nanos, Larry Hurtado, Pamela Eisenbaum, Geza Vermes, Amy Jill-Levine, and others since the third "quest for the historical Jesus" have been reassessing Jesus, Paul and the rest of the early church in its Jewish context, and finding that it fits better within that mold than the sort of syncretic views of the turn of the previous century. As far as later Christian legend is concerned, I can't really help you there. There's the obvious St. Thomas evangelizing to India stuff, but again, that might be better directed towards the /r/AcademicBiblical crowd.

In short, nothing. While absence of evidence is not evidence of absence, in this case I think we can largely dismiss claims of direct Indian influence on Christian thought. That is not to say there haven't been some interesting links, perhaps the most discussed being the Barlaam and Josephat story, a Christianized version of the mythic life of the Buddha.

If we have the same book, from what I can tell, Garbe is quite dismissive of Indian influence, however I will admit that I used a translation software, as I don't speak German. Still, he does call out a few other scholars, whose work I have been able to dig up. It's fascinating stuff, no doubt, but whether it has historical merit is less clear.

Let's start with the beginning of all this. The first time India was brought up with regards to Christianity was probably the Acts of Thomas. India was known to the Roman world, it was an important trading centre and, while their customs were strange to them, they had a fairly healthy relationship right up until the sack of Rome. So, in the 4th century a document appeared called the Acts of Thomas which alleged that Thomas, an apostle, went to India (well, mostly Parthia), and converted a king and his people before dying in South India. The story is fairly late, and most scholars doubt it's authenticity. Notably, it does contain some gnostic themes and focuses on the divine aspect of Jesus, rather different from the standard trinitarian. Anyway, I've discussed Christianity in India before here.

The British control of South Asia and the effects of colonialism were more two-sided than some have stated, one only needs to look at British cuisine to realize that. Part of this opening of influences was the examination of new avenues of scholarship related to the new colonies. The first major forays into this were in the field of linguistics and the construction of the proto-indo-european hypothesis. The earliest suggestions of Indian Influence on Christ that I can find was Louis Jacolliot’s 1869 Hindoo Origin of Hebrew and Christian Revelation. Half-theological treatise, half-historical treatment, it’s interesting, but it’s also drivel. Jacolliot appears to advocate the long disproven, though at his time contemporary, “out of India” theory of origin for the Indo-European people, taking Sanskrit as the root of Indo-European Languages. Out of the context of its era, it reads a little like the manifesto of a mentally ill conspiracy theorist, and even within context, it is a little strange. He actually suggests that Kṛṣṇa is actually Jesus, not that the legends influenced one another, but that they are the same person, which is historically iffy not for the least reason that one of them probably didn’t exist. For a more grounded, but no more probable, Unknown Life of Jesus Christ published in 1894. Notovich claimed to have found a lost manuscript detailing Jesus’ time spent as a student of Pāḷi and Sanskrit to Buddhist Monks. It was very quickly revealed as a hoax through the efforts of Indologists and even the Chief Lama of the Monastery he was alleged to have visited. Nonetheless, the theory has remained popular and is taken very seriously by some, especially the Ahmadiyya Muslim Community who see a continuity between world prophets.

Anyway, that’s the core of the arguments that I would consider completely out there. There are a few other ones which I should probably address. The first one is that there was a small Buddhist community in the Roman Empire which might have been part of the religious milieu of the East. This is unlikely, but not impossible. Indian traders came by both land and sea to Rome, and Jerusalem was hardly some backwater. But I would say that this is probably another fiction, albiet one that takes more consideration to actual history. Another is that Buddhist merchants and ideas filtered along the silk road may have influenced Jesus and the early Christians. I don’t think this is impossible, but the only research I’ve seen on it is fairly outdated and reaches pretty far. Once we get into Gnosticism and Apocrypha, I think there is more of a case to be made. Much of this literature emerged in the Syrian parts of the world, where Christianity, Judaism, Zoroastianism, Buddhism, and later Manichaeism all existed together. Manī, the founder of Manichaeism, which was a sort of Gnostic sect that took influence from Buddhism, went to Parthia and Kushana territory where he picked up some Buddhist ideas. Other Gnostic sects exhibit some possible influence from India, notably Jain style atomism and ascetic theory as well as Buddhist monastic styles. However, mainstream Christianity seems largely exempt from this, with one notable exception.

Awhile ago, I was doing research for a question about Manichaeism and ran into a paper called “Manichaeism and Augustine’s Idea of Massa Perditionis’” and ran into a familiar term. Essentially, Augustine’s concept of Massa resembles the Manichaean concept, which may originate with the Buddhist or Jain tradition or Skhanda. I feel uncomfortable speculating further, as I do not speak Syriac or Sogdian, thus am unable to read the possible Manichaean intermediary, and there is not a considerable body of research regarding Buddhist and especially Jain influence on Manichaeism. However, I will posit a possible influence of Buddhism and Jainism on Classical Christianity through Gnostic and Manichaean intermediaries.