Why did Europeans seek sea routes to India?

There are two reasons for this.  First, Europeans wanted to go to India because India was a major trade hub for spices, silks, and ivory--things that brought a hefty price at market in Europe.  Marco Polo started this in the thirteenth century when he wrote a travelogue about the Silk Road.  The second thing that we should examine is why Europeans needed a sea route to India. By 1500, the Ottoman Turks had taken control of the land route to India.  Constantinople fell in 1453 and Turkish forces were waging war in the Balkans.  While the Turks would let trade caravans through, they had to pay heavy fees.  Also, robbers from Muslim armies and Christian deserters would raid the caravans.  The sea route was supposed to be shorter according to the ancient Greeks, and one would bypass the deserts and mountains along the Silk Road that would be hazardous no matter who was in charge of the route.  

Why did Europeans seek a new sea route to India?

The Portuguese traveled around the tip of Africa in order to reach India and avoid Muslim traders and Venetian middlemen while still cashing in on the lucrative spice trade. Before 1500, most European traders who did business with Asia were willing to allow Venetian traders a cut of the business, as they were willing to do business with the Muslim traders of North Africa. If one was a European Christian and wanted to travel overland in order to gain spices, one could travel through Byzantium and not have to worry about dealing with the Muslim influence in the Middle East and North Africa.

After 1453, with the fall of Byzantium, this route largely closed as the Ottoman Empire fined any non-Muslim trader moving through the area. While the trip around the Cape of Good Hope was quite dangerous and time-consuming, the spices brought back from India made the survivors of the journey very rich. India became a place for the spices and goods of Asia to come together, and many cities such as Calcutta became world-famous for their trade goods.

Why did Europeans seek a western route to India?

Europeans were interested in a Western route to Asia because they wanted direct access to trade from India, China, and the spice islands in today's Indonesia. Previously, Europe's trade with the region was indirect, and controlled by a number of "middle men" culminating with Middle Eastern and Venetian merchants. Many emerging European nation-states realized that direct access to the trade could be highly profitable. So Columbus' belief that the riches of Asia could be accessed by sailing west were of interest to the monarchs of Spain. Of course, he referred to the people he encountered in the Americas as "Indians" in the mistaken belief that he had reached that region. Ironically, successful efforts to gain direct access to India and Asia were achieved not by sailing west, but east. Just six years after Columbus made landfall in the Bahamas, Vasco da Gama entered Calicut on the Indian subcontinent, the culmination of years of exploration around the tip of Africa and into the Indian Ocean. But the motivation for these explorers was to gain access to spices and other riches.

Why were European merchants interested in finding a sea route to Asia?

The merchants of Europe were interested in establishing better and more direct trade routes with Asia. Their interest in doing this is one reason why Europe turned outward during the Age of Discovery (circa the mid-fifteenth to the mid-sixteenth century), but it is not the only reason. Religious zeal, nationalism, and enhanced navigation techniques all contributed to the Age of Discovery.

Trade between Europe and Asia dates back to the Silk Road. Its name was appropriate because silk was the key export from Asia to Europe. The Silk Road transmitted ideas, diseases, and travelers (e.g., Marco Polo) as well as goods. The Mongol Empire, the largest in history, facilitated the use of the Silk Road. Merchants relied on the Silk Road to earn their fortunes.

By 1400, however, the Mongol Empire had fallen apart. In its place, a hostile Ottoman Empire emerged. The Ottoman Empire conquered Constantinople in 1453, ending the Byzantine Empire. Venice also controlled important trade routes. European merchants wanted to have their own routes so that they would not have to rely on belligerent Ottomans and avaricious Venetians.

After Constantinople had fallen in 1453, two European nations, the monarchies of Portugal and Spain, were prepared to undertake ambitious sea voyages. This support by monarchs was crucial because merchants lacked the means to launch such enormous and costly voyages. Portugal was especially interested in the African gold trade and—ultimately—a sea route to India. Christopher Columbus sailed West in an effort to reach Asia.