Is it important for us to understand the culture during the time the New Testament was written? Let me give examples of why it is important.
First, we must understand that people of different races did not associate with one another back then and women were treated like second class citizens. So understanding this, a greater impact is made knowing Jesus, a Jew, asked a Samaritan woman for a drink of water at the well. He not only crossed racial boundaries, that were wider in their culture than today, but he also stretched out his hand to a woman.
Another example of why we need to understand the culture then is when the woman, who was caught in adultery, was brought before Jesus. Back then, adultery was a crime punishable by death, whereas today, it is overlooked by society. The woman was brought before Jesus, more as a test than anything else, but He looked upon her with mercy and love instead of judgment. If a woman is found in an adulterous relationship today, it is hardly noticed let alone considered as a crime punishable by death. Knowing this was a death sentence for her then makes a greater impact when Jesus showed her mercy and love when He said “Go and sin no more”.
The language spoken then and the original language the Bible was written in does not always translate to English perfectly. Therefore, we should understand that some words meant something different back then. This knowledge can mean the reader understands the Scripture differently and may have a truer understanding of the Word of God.
If we were to look at the New Testament in light of today’s culture in America, many truths and lessons Jesus taught would be overlooked and lost. So yes, it is vitally important that we understand the culture during which time the New Testament was written in order to better understand His teachings.

Response 1:
I agree completely with your