How should a Christian deal with destructive higher criticism of the Bible?
As a believer who also happens to be a rhetorician, about the only element I can add to River C's and Narnian's answers is the importance of the role of rhetoric in determining which side of the fence one is on regarding any controversial issue in general, and the issue of higher criticism in particular. In a nutshell, the role of rhetoric is significant in critical and often overlooked ways.
First, there is nothing wrong with being fully persuaded of one's point of view. Just as the Apostle Paul was fully persuaded that God was able to keep whatever Paul had committed unto Him until the day of Christ's return (see 2 Timothy 1:12), we too should be unashamed to be fully persuaded there is indeed something special about the Bible in both its historicity and the authoritative nature of its claims in all things pertaining to life and godliness.
Just as an aside, which probably resembles more of a criticism of higher critics than an aside, I suggest there are more godly people on the side of those who believe in the inspiration of the Scripture than on the side of those who believe the Bible cannot be taken at face value. That alone is significant, I feel. With that said, let us proceed.
Second, persuasion, if it is truly heartfelt and ethical, and not simply conjured up to take advantage of people, requires some bedrock assumptions or presuppositions. If it lacks them, persuasion could not exist, since to persuade someone of something, one needs to have been persuaded of that something in the first place, unless of course one is simply posturing, for whatever reason(s).
Think about it. Even a purely informational type of speech, whether it concerns how to fix a kitchen faucet or how to construct a nuclear weapon, is grounded in certain assumptions, not the least of which is that words when strung together actually make sense and have real-world implications.
In the English language, for example, the words Allen wrench and Allen screw correspond to a common tool and part, respectively, that are used in repairing some kitchen faucets. Not so coincidentally, the right sized wrench fits the corresponding axial hexagonal hole in the screw's head perfectly, whether to tighten it or to loosen it! If the speaker who is imparting this crucial information were not fully convinced of this fact, he would not be speaking in the first place, unless of course the faucet he's informing us how to repair does not contain an Allen screw! But I digress.
The point is, persuasion is at the heart of any pronouncement, whether it is overtly persuasive, informative, inspirational, or legal, or not. We commonly call the form this persuasion takes assumptions or presuppositions. Fortunately or unfortunately, depending on one's perspective, these presuppositions are often taken for granted. Personally, I think this is a good thing, since in virtually any communication, written or spoken, it's nice not to have to re-invent the wheel, so to speak, every time you open your mouth or commit some words to paper!
For example, in discussing the Bible with fellow believers, I need not defend my use of the word "Paul" whenever I quote a verse from one of his letters in the New Testament. Chances are better than good that my peers assume the words I quote to have originated in the Apostle Paul and that whether or not they actually came from Paul is in a sense immaterial or moot, since the words are authoritative anyway!
In discussing the Bible with antagonistic unbelievers, I need to take a completely different tack, since we do not likely share the same assumptions about the authoritativeness of that verse from Paul's letter. I would then need to make my presuppositions clear at that point and hope that my interlocutors would do the same. Chances are, we would likely need to agree to disagree agreeably, at least on my part. After all, I'm the Christian, right?!
This is not to say that it would be unseemly for me to engage in good-natured debate with my interlocutors and appeal to their rational faculties by delving into bibliographical proofs (i.e., the existing manuscript evidence for the historicity and accuracy of the Bible); internal and external evidences as to the people, places, things, events, and dates contained in the manuscripts of both the Bible and non-biblical sources as well; and anecdotal evidence associated with people whose lives have been transformed, sometimes radically, by believing the Bible is in fact God's word and not some cobbled-together hodgepodge of historical, hortative, and histrionic elements.
In other words, you cannot really talk about presuppositions without getting into apologetics, in which both unbelievers and believers need to be conversant if they want to be taken at least somewhat seriously. I am not advocating, however, that Christians debate the higher critics, for example, by attempting deliberately to embarrass and/or shame them. First Peter 3:15 makes it perfectly clear that Christians should defend their faith by
By reverencing Christ in our hearts (i.e., asking ourselves, "What would Jesus do or say in this situation?")
By being ready to mount a defense (or in Greek, apologia)
By answering everyone who asks, not just nice people but not-so-nice people, too
By giving an account of the hope we have in Christ
By doing all the above in a spirit of gentleness and reverence, and by keeping a clear conscience, so as not to become a babbling hypocrite (see v.16)
In summary, as people of the Word we need to love the higher critics; we need to debate them not only to persuade them of our point of view but to learn from them humbly and to commend them sincerely for whatever legitimate insights they provide; and we need to pray for them and be ready to defend our faith and at the same time realize we cannot convert anyone, since that is the Lord's work and not ours. Our job is always to be ready to engage people in considering the claims of Christ on their lives. To that end, a drop of honey beats a gallon of gall, or to quote the Apostle Paul, who may or may not have written(!) Colossians 4:6,
"Let your speech always be with grace, as though seasoned with salt, so that you will know how you should respond to each person."