Question

What did The Europeans hope to gain by finding a sea route to asia

Answer 1

Basically they wanted to cut out the middlemen. Arabs were in the middle between India, China, and Europe. Europe was paying very high prices for spices and other goods. Islam and Christian Europe were not friendly towards each other. The Portuguese were actually looking for a long lost Christian sect they believed lived in the far east when the discovered how to sail to India and thought that Hinduism was a Christian religion at first when they arrived in India.  

Answer 2

Trade was the motivating factor. The silk road accross Asia was besieged by bandits & required bribes to a number of nations to transport goods to & from Asia. Therefore a sea route would be a much safer & less expensive means of getting trade goods to & from Asia.