It is often believed that the "wine" spoken of in the New Testament, is more comparable 
to our grape juice now a days. 
 
Here is a reference for that assumption: 
 
"Elder Bruce R. McConkie wrote that the wine spoken of in John 2:1-11 was a drink made 
from the "fruit of the vine" (Matt. 26:29), a light, sweet wine (normally unfermented); 
eaten with bread it was one of the staple foods of the day. (Doctrinal New Testament 
Commentary, 1:136)."

in addition to phillip's comment, wine back then was not strong at all. i had very little alchohol so was very similar to our grape juice today 
hope that clears it up :)

Also we must remember that the word of wisdom is not an ancient commandment. In the ancient days there was a law of health specific to the problems of the day. The commandments in the law of Moses are now known as the Kosher laws and helped the ancient Hebrews to survive and live healthier lives thousands of years ago. In the second verse of D&C section 89 (where the LDS law of health known as the word of wisdom comes from) it reads "showing forth the order and will of God in the temporal salvation of all saints in the last days." The important words for this conversation are "the last days." This law was not in effect during the time of the Bible nor the Book of Mormon. It would not have made sense back then but was given in 1832 for "the last days" to specifically help us in our time to live healthier lives.

I just want to add the fact that in "biblical times", water was not commonly drunken due to the fact that they didn't know how to purify it like we do today, and drinking wine was safer, even with the slight fermentation it went through.