A recent version of this question [labelled a duplicate] asked how Christians should respond the claim. This is different from "What evidence is there that he did or did not visit India?"

My answer is that Christians should respond by affirming that it's not out of the realm of possibility. As others have pointed out, there is no hard evidence for it but that does not mean it could not have happened. A mutually respectful dialog on the issue is also an opportunity to testify to one's own faith. I doubt that most Christians' faith depends on whether Jesus did or did not go to India. In theological terms the issue is a "theologumenon", an issue on which Christian may disagree without either party falling into heresy. So the answer to the question "how should we respond?" is: look at it as an invitation to dialogue, a perfect "teaching moment."

There is no biblical support or any evidence that Jesus left the land of His birth at any time, leave aside the speculation that He was in India before beginning His ministry in Israel. From the time when Jesus lived, it is unreasonable to expect to have each and every details account of any person, like we can have it today on any medium. Of the four Gospel accounts, only two mention the birth of Jesus (Matthew and Luke), and only one (Luke) mentions anything about Jesus' life prior to His beginning His three-year ministry in Israel.

While the Bible doesn't explicitly say this, it is implied from the following passage in the Gospel of Luke that Jesus grew up in Nazareth with His family until it was time to begin His ministry:

Luke 4:16,22-24

He went to Nazareth, where he had been brought up, and on the Sabbath day he went into the synagogue, as was His custom. And he stood up to read.... All spoke well of him and were amazed at the gracious words that came from His lips. ‘Isn't this Joseph's son?’ they asked. Jesus said to them, ‘Surely you will quote this proverb to me: “Physician, heal yourself! Do here in your hometown what we have heard that you did in Capernaum." I tell you the truth,’ he continued, ‘no prophet is accepted in his hometown’”.

Notice that Luke says that Jesus was "brought up" in Nazareth, and he also mentions twice that Nazareth was Jesus' hometown. Furthermore, the people in the synagogue knew Jesus and knew that he was Joseph's son. All of this leads to the conclusion that Jesus lived in relative obscurity in Nazareth until His baptism.

Despite this fairly clear account, there are those who want to fill in the gaps in Jesus' life with extraordinary tales of adventure and mystery. Their ultimate source is the father of lies, Satan (John 8:44). As he did in the beginning, so he is doing now. At the baptism of Jesus, a voice from heaven proclaimed,

Matt.3:17 “This is my Son, whom I love; with him I am well pleased”.

Jesus of Nazareth was declared by God the Father to be His one and only Son. All of these conspiracy theories attempt to divert us from God's declaration that Jesus was His Son. They do so by downplaying or denying outright the deity of Jesus Christ. By denying His divinity, they reduce Jesus to just another rabbi, prophet, sage or wise man.

If Jesus went to India prior to His three-year ministry, then one would expect there to be a distinct Indian flavour to His teaching. However, Jesus' displays vast knowledge of the Torah. Jesus quotes the Hebrew Scriptures all throughout His earthly ministry to the point of correcting even the learned scholars of His day. His teaching style was consistent with the Jewish itinerant teachers of His day.

Finally, if we assume for the argument sake that He came to India and studied some things about India then one would wonder why he ignored and not mentioned any Indian names while mentioning all the prophets of one true God.

Where Jesus was during those times cannot be said with any academic certainty. No credible source ever even mentions it. Quite a few of the other answers point out exactly why Jesus was very likely never in India ¹, so I will neglect to parrot it.

I do want to point out one thing that I have learned recently. A contemporary of Jesus was Apollonius of Tyana. Not only are there similarities between them concerning claims of miracles, including resurrection, there are also similarities concerning teachings. The more I look into Apollonius the more I see subtle similarities in their teachings. Whether Apollonius' or Jesus' stories borrow from each other is really a matter of opinion, however, we would be foolish to not think that there might be some. On the unsupported point that Jesus was in India, we can argue that it was borrowed from the stories about Apollonius because he certainly was in India for several years. Considering some of their similarities, it is not a stretch to think that the first person who mentioned that Jesus was in India was actually referring to Apollonius.

Just wanting to finish the thoughts, Jesus and Apollonius did have several major differences in their lives. The first is that Apollonius did travel during his "ministry", visiting a range as large as Rome to India. Jesus, relatively speaking, did not. In addition to that, Jesus rarely went into the large towns. Apollonius focused almost exclusively on them. The second is that Jesus was a constant critic of the leaders and wealthy; Apollonius was a friend to them.

1: When I say India I mean Jesus' geographical, contemporary equivalent, who's name eludes me at this time.

Jesus was called Rabbi (Teacher of the Law of Moses) by the Jews.

John 3:1-2 (NIV) Now there was a Pharisee, a man named Nicodemus who was a member of the Jewish ruling council. He came to Jesus at night and said, “Rabbi, we know that you are a teacher who has come from God. For no one could perform the signs you are doing if God were not with him.”

To be called a Rabbi, one must be very expert in Torah. People spend their entire life learning the Torah. It is vast and complicated. Jesus was very expert in Torah and no one at that time could argue with Him. Jesus always quoted from the Old Testament, I can't list them all here.

If Jesus was expert in Torah, it means that He spent his 12-30 years learning the Torah, if Jesus needed to learn something. Torah was available only in the hands of Jews, not in India. But I doubt that Jesus needed to learn because He is the Word of God himself, He is the creator of all things, His wisdom is far beyond our reach. He didn't have to learn from His own creation. Instead, He came to teach us about God, He came to lead us to God.

If you look at the world map, India is very far away from Israel and there is no direct sea route. If Jesus needed to learn something more apart from the Torah, He could have gone to Greece to learn from the great philosophers and mathematicians out there. Greece was much closer and could be reached by sea voyage.

Here is an example of Jesus replying from the Torah when a man asked about inheriting eternal life.

Luke 18:18-22 (NIV)

A certain ruler asked him, “Good teacher, what must I do to inherit eternal life?” “Why do you call me good?” Jesus answered. “No one is good—except God alone. You know the commandments: ‘You shall not commit adultery, you shall not murder, you shall not steal, you shall not give false testimony, honor your father and mother.’” “All these I have kept since I was a boy,” he said. When Jesus heard this, he said to him, “You still lack one thing. Sell everything you have and give to the poor, and you will have treasure in heaven. Then come, follow me.”

Jesus did not explain the man about Samsara (endless rebirth) and how to attain Moksha (liberation from endless rebirth), instead, Jesus told the man to sell everything and follow Him because there is eternal life in Him (John 3:16).

Where was Jesus from his age 12 to 33?

From age 12 to 30 he was in Nazareth. And from age 30 to 33 he was in and around Jerusalem. Where else? And no, he was not in India.

Why does scripture not mention this period in Jesus' life?

Because scriptures are not biographies of Christ. Their intention and purpose was different. They wanted to record the story of salvation. And from a Christian perspective, by divine providence, the hidden life of Christ (age 12 to 30) is not necessary for that.

India as a hub of knowledge

If assuming that Jesus traveled to gain knowledge, rather than India, a more convenient and feasible destination would be Rome. Rome was connected by road to Jerusalem and by extension to Nazareth too. Think about the dangers of crossing all those countries to reach India. Also, India was not a single country at that time. A host of kingdoms ruled over India in those days. (All of this is under the assumption of ignoring Jerusalem, which was the center for Jewish studies, and the assumption that Jesus needed to travel to gain knowledge, which he never preached about or talked about in his later years.)

Other problems with this hypothesis

There are many other problems in this hypothesis. Some of them are:

1. There is absolutely no concrete evidence. Absolutely none!! Those people who claim that he went to India say that there are scrolls hidden deep in some monastery. But that is not evidence. It is just fables. Anyone could claim anything like this. These are not mentioned in any reliable sources, nor can they be verified at all.
2. Being a devout Jew, Jesus would not have studied in a polytheistic religion. If he had, at least it should have been reflected in his teaching, which it is not.
3. We know from his public life that just like every other devout Jew, Jesus made a pilgrimage to Jerusalem every year. The last recorded event of Jesus' childhood was a pilgrimage to Jerusalem. From this we can reasonably assume that he made the pilgrimage during his hidden life too. Even a single trip to and from India that time would be extremely dangerous and difficult no matter what the means of transportation. Also, the trip would be very costly, and Jesus' family was poor.
4. Jesus was known in his community as a carpenter, since it was a Jewish tradition for young boys to learn their father's trade. His parables were even influenced by his job. He told of building a house on a rock as opposed to sand (Matthew 7:24-27). This is a great indication that Jesus has spent his adolescent years in his own community.
5. There are numerous passages in the Bible stating that people recognized him and refused to believe in him because he was just a carpenter and they knew him too well to accept him. This implies that Jesus was very well known in his community, which would not have been possible if he had been in India for most of his life. (See Matthew 2:23; 13:54-56; Luke 2:51-52; 4:22.)
6. Why did Jesus not talk about India to his disciples? None of the early records mention such a journey. The Jewish leaders did not like Jesus. So when they condemned him, they would certainly have said that he had learned heretical things in a foreign land. They knew who he was, and the people knew who he was. So this going to India hypothesis does not hold water.

There is a similar theory called the Swoon hypothesis, which claims that Jesus went to India after his death but before the ascension. Covering this would be beyond the scope of the question, but Wikipedia does a decent job of explaining the problems with it.