How should a Christian deal with destructive higher criticism of the Bible? In recent times authorship of Biblical books has been called into question by modern scholars.

It is believed that Isaiah didn't write Isaiah, John didn't write John, Matthew didn't write Matthew, and so on. A little concession is given to us by conceding that Mark did write Mark, but he wasn't an eyewitness so his testimony doesn't matter anyway.

My question is - 


How much weight does this kind of textual criticism have? What logic/principles are used to arrive at such kind of results? (Can you give an example?)
What should be the Christian response to such claims? What can a Christian do to educate herself on these matters?
You need to deal with them on a case by case basis. For one, it is helpful to understand that all claims of authorship are always uncertain, even and especially those made by textual critics. As Christians we accept a tradition about who wrote the books along with whatever authorship may be claimed therein.

In the case of the Gospel writers, we can take note of two problems in 'identifying' the author using textual criticism:


The authors shared material and did not have a notion of 'plagiarism' - so some stuff may have been shared directly. Additionally there is believed to have been a 'Document Q' which was an early record of many of Jesus' sayings written down by his disciples. It did not survive though most or all of it may actually just be in Matthew, Mark and Luke. This could make analyzing the text for style and origin difficult and subject to a lot noise that would make precise identification impossible.
In the case of John, it is traditional that he did not write his Gospel (or Revelation) by hand but in the prior case, the witness - the Gospel - is John's account. The end of the Gospel communicates this, not necessarily that John is sitting there writing (he was old at the time, it is believed that Prochoros was the actual writer) but that he is the witness of these things. With Revelation we know for sure he was having the vision and telling Prochoros about it, who was writing it down. So the result would be a mixture of John and Prochoros, not John alone. 


With Hebrews, Paul is traditionally accepted as the writer. Textual critics, in their attempt to identify authorship, may forget that an extremely learned man such as Paul was fully capable of concealing his identity in his writing, as well as utilizing different styles for various purposes. Perhaps they assume that ancient people were all non-self conscious and didn't have the ability to choose words and phrases that were different than their normal style or pattern for some purpose of their own.

For Isaiah, we must remember again the problem of scribes - so to say Isaiah didn't 'write' Isaiah doesn't say much! How could a prophet write his vision while having it? So with Isaiah we may easily have a collection of works from different eras in Isaiah's life written down by different scribes. Trying to be exact about it simply makes it less clear.

In general, Christians must remember that 'Sola Scriptura' is a recent invention and is not the basis for traditional faith. Even the original holders of Sola Scriptura held to various oral and written traditions about the scriptures - such as the ever-virginity of Mary - that are not directly witnessed by the scriptures themselves.

Consider the case of the books of Moses - given the potential age of Genesis, Exodus, Leviticus, Numbers and Deuteronomy - plus the history itself witnessed by the scripture - we cannot be fully certain (ever) that what we have is word-for-word of what Moses either dictated or wrote. Our trust is in the witness of the Spirit, not in the inerrant words of this or that human (guess what, no humans are inerrant.) If you learn a bit about how human beings interpret visionary works (such as prophecy) it becomes clear that perfect word-for-word accuracy does not necessarily mean accuracy on many points. 

Thus it becomes nearly impossible to debunk the scripture - all attempts have failed really - because on the points on which our faith hangs, such as the divinity of the Son of God, the incarnation of Him, his death, resurrection, ascension and sending of the Holy Spirit, the witness doesn't vary. It is possible that Genesis 3 was added later during the Babylonian captivity (Genesis certainly reflects two different writings, but whether they come from the same person or not cannot be established) but whether it did or not doesn't matter. One reason for this idea is that it attempts to undermine the Christian belief in the fall. But consider how it fails to do so: If it was added in the Babylonian times, that indicates that the teaching of the Fall of Man has been in the people of God since before then, and it is also consistent with other teachings in the scripture. 

Textual Criticism - inasmuch as it is a tool to undermine Christian faith - can really only undermine faith based on literal scriptural inerrancy. Those who hold to a tradition (kergyma) about scripture may or may not find such things interesting, depending on their background and talents, but they do not pose a challenge to the faith, since the faith is not just the Bible, but the people (the Church) who have been witnesses of God since the beginning of time. Even if the fall part of Genesis is a later addition, it is consistent with the teaching of the Church and is thus part of the faith anyway. Those 'additions' if they were, were just as inspired as the text they added to. 

It also helps to remember that nobody is going to rediscover the 'real faith' of the Bible at any time. It still exists and continues and most attempts to radically reinterpret the scripture are, as they have been since the time of Christ, the work of Gnostics.