Some traditions state that Jesus visited India between the ages of 12 and 33, in order to study and learn. What evidence is there that he did or did not visit India?
Where was Jesus from his age 12 to 33? Was he in India for study? Else where was he?
From age 12 to 30 in Nazareth. And from 30 to 33 in and around Jerusalem. Where else? And no he was not in India.
Why most scriptures are silent on this?
Because scriptures are NOT the biography of Christ. Their intention and purpose were 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.
Before 2000 years ago, India was the destination of the world to get the knowledge(and now also), I believe that chances are high on this case?
If assuming that Jesus traveled to gain knowledge, I am sorry, rather than India, 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 remember India was not a single country at that time, a host of kingdoms ruled over India in those days. (All of this in the assumption of ignoring Jerusalem, which the center for Jewish studies, and that Jesus needed to travel to gain knowledge, which he never preached about or talked about in his later years.)
There are many other problems in this hypothesis. Some of them are:
P.S: There is another one called swoon hypothesis which claims that he came after his death. Explaining that would be out of scope for the question you have asked. But I guess Wikipedia does a decent job in explaining the problems with it.
Jesus was called Rabbi (Teacher of the Law of Moses) by the Jews.
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.
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).
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:
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,
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 may very well have went to (pre)India. The complete silence about it being precisely because Jews at the time would have regarded uttering Indian teachings to be criminal heresy. He may have been so moved by what he learned that he dressed the lessons he was passing on in an orthodox manner, making them more palatable to the Jews of the time. Many of his thoughts and ideas were very much the same as Buddhists were teaching, and it would not have taken much effort to blend them with Pharisee theology. I promise, I am not the devil, but am implored to play devils advocate here as so many Christians have railed on your question with little more than religious bias as their backing. Explore the idea, seek out evidence. Because if it did happen and it was never known simply because of peoples assumptions, it would be an unimaginable tragedy.