Claims of the St. Thomas Christians are very old, dating back to at least the 3rd century, but they also aren't as old as St. Thomas, which means they probably aren't true. What is true, however, is the presence of Christians on the edge of India since ancient times.

But lets get hypothetical. We have not direct evidence of a Christian Presence in India before the 3rd century, all we have are the Acts of Thomas and the Songs of Thomas. These give a small picture of Christianity in the first century but are hardly reliable. However, the Songs of Thomas seem to be older than Syrian Christianity and related to Keralan Jews, rather than Syrian Worship, whether that means that they were converted Jewish Keralans or they simply adopted customs is unknown. Christians in India are also mentioned in the texts of Manichaeism, though only in the hagiography of Mani which isn't exactly reliable, and on the far edges of India in Kushan territory. However, this could mean that there was a small community of Christians in ancient India.

The first Christians in India that we have reliable evidence of were of Persian Import, being part of the Church of the East. Initially a group of merchants in Kerala, they eventually integrated with the local population and established their own church. It is possible they also adopted some of the local customs, as the Portuguese mention Brahminical practices when they encountered them. They had a thriving culture and established many churches in the nine centuries between establishment and re-contact with the west (it was severed in during the Muslim conquest of Persia).

The Church in its present form has more to do with the Portuguese and the British than with any earlier efforts. The Portuguese set about growing and reviving the small community of Christians in India when they first encountered them and removing any Indian influences from practice. This also served as an important economic means, since they would far more easily establish a foothold if they could identify with the locals. The built a church over the ruins of a sixth century Syriac one and designated it as the Church of Thomas, where they supposedly found his relics. That said, the myth was already in some circulation then, but it was the Portuguese who identified them as such (though they also identified the Vaiśnavists as Christian, so yeah). The British continued their efforts and rebuilt the church in the 19th century.

In any case, it isn't likely that the story is exactly true. If there was a Thomas who went to India, it is more likely he converted the Jewish population rather than the Brahminical one, or perhaps got them to accept Jesus within their tradition, but I doubt the miraculous conversion of thousands happened. However, there was an ancient community of Christians in Kerala dating back to at least the sixth century and probably as early as the third or fourth. It is impossible to entirely discount the idea that St. Thomas went to India and established a community of Christians there, but there is little proof.

• Malekandathil, Pius (2010) Maritime India

• Frykenberg, R.E. (2008) Christianity in India

I would like to add a bit of background information to the good answer by /u/artfulorpheus.

Before the days of emperor Constantine, the veneration of Christian martyrs was already a centerpiece of Christian worship. Christians would worship at the tombs of those great examples of the faith, those who died suffering like Christ in His name. During the reign of Constantine, the emperor had churches built over some of those tombs to create Christian places of worship for a suddenly fast growing population of Christians. The basilicas of St Peter and St Paul were built over the burial places of those two apostles, not far outside of the city of Rome. The basilica was originally the Roman building that the emperor would use as a meeting hall, with his throne at the apse end of the long nave, with extra room on the two side isles. The Christian basilica became the iconic place of worship and the standard building type up to the modern day. Round churches based on the Roman martyrium were built in Rome (Santa Costanza in Rome, also quite a bit outside the walled city of the late empire) and in Jerusalem (the Holy Sepulcher) over the site of the supposed entombment of Christ.

The constantinian basilicas were built with the apse over the tomb of the saint. Having a tomb with an important saint to build a church over it was essentially a requirement for a religion that venerated those early Christian martyrs and apostles. But the legality of Christianity after the Edict of Milan meant that it was unlikely that a community that did not already have a well-known martyr would acquire one in the future to be able to claim what was needed for a church building of renown in the Christian world. This is the time when we start to see apocryphal lives of saints (hagiographies) that often claim that apostles traveled to faraway places to convert the local population. The travels of the apostle Paul narrated in Acts made for a template for the unlikely stories attributed to some of the other apostles. The story of Thomas is very likely to be one of these unlikely stories that just happened to place an apostle in a Christian community of the 3rd-5th century at the time of his death.

And very soon they ran out of the original apostles to claim for your church, and that is when we start to see the idea of relics, taking the bones of saints and martyrs from their tombs and transporting them to the altars of a new church, nearby or even a long distance away, to be venerated at the altar. That was something that was initially very controversial and opposed by some because of the idea that saints would need those bones at the end of days to take them along to heaven. And eventually those relics get their individual containers, bejeweled reliquaries, very much transportable, that becomes one of the main objects for Christian worship and medieval Christian art.

In general, the idea of first century apostles traveling to places far outside the then core region of Christianity is a later invention associated with the need to claim important burials for worship and church building.