Westerners were interested in Asia mainly for economic reasons. They wanted to tap the resources of the continent to get rich. Later, they also wanted Asia for religious reasons (convert them to Christianity) and military reasons (have bases to get more military power).
What impact did that interested eventually have? That depends on how long of a time frame you look at. If you take it all the way up to the present, the impacts include World War II and the current rise of China as a major power.
When the US "opened" Japan, for example, that caused Japan to try to become like the West. After the Meiji Restoration, Japan became a major power. This led to World War II.
The proverbial “three g’s”—gold, God, and glory—were the primary motivators for the West's interest in Asia. Religious passions inspired the crusaders to initiate their quest to retake the Holy Lands. More to the point, the Jesuits and other missionary orders established proselytizing enterprises throughout Asia. In the Philippines, Spanish missionaries introduced Christianity, often at sword point, while elite-oriented missionary endeavors by Matteo Ricci and his brethren sought converts in China.
The lure of fortune to be made in the Asian spice, tea, silk, or various other trades attracted not just the attention of merchants but of entire Western nation governments. Europeans quickly discovered that whoever dominated the seas and their routes to Asia could monopolize that trade over the entire European market. Thus, despite their tenacious attempts to maintain their monopoly, the Portuguese lost control to the Dutch, who in turn could not hold back the British.
Finally, national glory and honor came to be associated with the strength of one’s position in Asia. Part of that glory came in the form of wealth derived from the Asian trade, and part from displays of the naval power required to maintain dominance on the high seas. The attractiveness of Asia and the benefits mentioned above led explorers, such as Columbus himself, to discover new lands and Europe’s eventual colonization of most of the globe, either by claiming lands discovered while trying to get to Asia, building on lands en route to Asia, or establishing footholds in Asia itself.