What do Evangelical Protestants say about sanctification?

If there's any one denomination finds itself consistently in accord with "broad evangelicalism," it's gotta be the Southern Baptist Convention. Certainly that's true on this issue; article 4 of the Baptist Faith and Message says:

In its broadest sense salvation includes regeneration, justification, sanctification, and glorification. [...] Sanctification is the experience, beginning in regeneration, by which the believer is set apart to God's purposes, and is enabled to progress toward moral and spiritual maturity through the presence and power of the Holy Spirit dwelling in him. Growth in grace should continue throughout the regenerate person's life.

Methodists/Wesleyans are also well-represented in the evangelical movement, and they would agree with what's quoted above, but they would add the concept of entire sanctification.

Lutherans too are well-represented in evangelicalism, and would probably sign on to the Baptist statement above. But they would emphasize strongly that justification and sanctification are separate and must be kept separate in preaching. Lutherans are big on what they call the law-gospel distinction, and the gospel of free/unmerited grace (the grace of imputed righteousness, resulting in justification) must not be mingled with the preaching of the law, which sanctifies the justified but condemns the unjust.

Then you've got Presbyterians (which is basically synonymous with reformed). Almost everything I said above about Lutherans applies to them as well, although they tend to approach the law-gospel distinction a little bit differently. Since I'm most familiar with them, and since they're perhaps the most confessionally verbose anyway, I'll quote two of their confessions, starting with the Westminster Confession of Faith (chapter 13). This pretty concisely defines the Protestant doctrine of sanctification. And if you don't read too much Presbyterian baggage into "effectual calling" in the first sentence, and if you discount the pieces that contradict Methodist "entire sanctification" teaching, it pretty well encapsulates a view of sanctification that any evangelical should be able to raise his glass to:

They, who are once effectually called, and regenerated, having a new heart, and a new spirit created in them, are further sanctified, really and personally, through the virtue of Christ's death and resurrection, by His Word and Spirit dwelling in them: the dominion of the whole body of sin is destroyed, and the several lusts thereof are more and more weakened and mortified; and they more and more quickened and strengthened in all saving graces, to the practice of true holiness, without which no man shall see the Lord.

This sanctification is throughout, in the whole man; yet imperfect in this life, there abiding still some remnants of corruption in every part; whence arises a continual and irreconcilable war, the flesh lusting against the Spirit, and the Spirit against the flesh.

In which war, although the remaining corruption, for a time, may much prevail; yet, through the continual supply of strength from the sanctifying Spirit of Christ, the regenerate part does overcome; and so, the saints grow in grace, perfecting holiness in the fear of God.

Now comes the Belgic Confession (article 24), which predates the Westminster Confession. It says much the same, but it mixes in some polemic specifically designed to parry Catholic attacks on the doctrine:

We believe that this true faith, produced in us by the hearing of God’s Word and by the work of the Holy Spirit, regenerates us and makes us new creatures, causing us to live a new life and freeing us from the slavery of sin. Therefore, far from making people cold toward living in a pious and holy way, this justifying faith, quite to the contrary, so works within them that apart from it they will never do a thing out of love for God but only out of love for themselves and fear of being condemned.

So then, it is impossible for this holy faith to be unfruitful in a human being, seeing that we do not speak of an empty faith but of what Scripture calls “faith working through love,” which moves people to do by themselves the works that God has commanded in the Word. These works, proceeding from the good root of faith, are good and acceptable to God, since they are all sanctified by God’s grace. Yet they do not count toward our justification— for by faith in Christ we are justified, even before we do good works. Otherwise they could not be good, any more than the fruit of a tree could be good if the tree is not good in the first place.

So then, we do good works, but not for merit— for what would we merit? Rather, we are indebted to God for the good works we do, and not God to us, since God “is at work in [us], enabling [us] both to will and to work for his good pleasure” — thus keeping in mind what is written: “When you have done all that you were ordered to do, say, ‘We are worthless slaves; we have done only what we ought to have done.’“ Yet we do not wish to deny that God rewards good works— but it is by grace that God crowns these gifts.

Moreover, although we do good works we do not base our salvation on them; for we cannot do any work that is not defiled by our flesh and also worthy of punishment. And even if we could point to one, memory of a single sin is enough for God to reject that work. So we would always be in doubt, tossed back and forth without any certainty, and our poor consciences would be tormented constantly if they did not rest on the merit of the suffering and death of our Savior.

From each of these, we could synthesize a statement on sanctification that most or all evangelicals would agree with:

We are able to be justified by virtue of Christ's death and resurrection. The justifying, unmerited grace of God frees us from selfish motives for seeking holiness. Sanctification is gradual, aided by the work of the Holy Spirit on our hearts. It comes entirely after we have already been justified.

Though justification and sanctification are separate, it's important to remember that the same faith that justifies also sanctifies. John Piper, a reformed Baptist, wrote a book with that premise with the aim of aiding Christians in their sanctification and in their joy. It's called Future Grace, and it's considered a modern classic.

For more comparison and contrast between the various Protestant views of sanctification, you might want to check out two books on five views of sanctification. Both of them document reformed, Wesleyan, and Pentecostal views. The former book adds Keswick and "Augustinian-Dispensational," and the latter adds Lutheran and contemplative. (Keswick and Pentecostal theology both are closely related to Wesleyanism.)