What is the biblical basis against Sola Fide (salvation by faith alone, apart from works)? One of the key points of the Reformation was the doctrine of Sola Fide: that salvation is by faith alone, apart from works. The Reformers thought this was in contrast to the doctrines of the Catholic church.

What, then, is the biblical basis against the doctrine of Sola Fide?
The question could be worded better. Even the reformers would disagree with Sola Fide as you've described it (with the exception of Luther, perhaps).
A common passage against Sola Fide is found in James 2:

14 What good is it, my brothers, if someone says he has faith but does not have works? Can that faith save him? 15  If a brother or sister is poorly clothed and lacking in daily food, 16  and one of you says to them, “Go in peace, be warmed and filled,” without giving them the things needed for the body, what good is that? 17 So also faith by itself, if it does not have works, is dead.
18 But someone will say, “You have faith and I have works.” Show me your faith apart from your works, and I will show you my faith by my works. 19  You believe that God is one; you do well. Even the demons believe—and shudder! 20 Do you want to be shown, you foolish person, that faith apart from works is useless? 21  Was not Abraham our father justified by works when he offered up his son Isaac on the altar? 22 You see that faith was active along with his works, and faith was completed by his works; 23 and the Scripture was fulfilled that says, “Abraham believed God, and it was counted to him as righteousness”—and he was called a friend of God. 24 You see that a person is justified by works and not by faith alone. 25 And in the same way was not also Rahab the prostitute justified by works when she received the messengers and sent them out by another way? 26 For as the body apart from the spirit is dead, so also faith apart from works is dead.

The problem is that there is more than one aspect of salvation. There is justification, that which is Sola Fide, and there is sanctification, which is accomplished by the Holy Spirit and is manifest by good works. This is why 2nd Peter tells us

10 Therefore, brothers, be all the more diligent to confirm your calling and election, for if you practice these qualities you will never fall.

So salvation is never Sola Fide. However, salvation is also never of works.