Is our theology sufficiently biblical to understand statements where God not only hates the sin, but also is said to hate the sinner (see, for example, Ps. 11:5-6)?

I think we can find some wise counsel in an article by John McKenzie written over 60 years ago. He writes:

There is a lawful hatred of the sinner; and indeed there must be, since such a hatred is the obverse of the love of God. The love of God hates all that is opposed to God; and sinners–not merely sin–are opposed to God. And if such a sentiment is lawful, its expression is lawful; and one may desire that the evil in another receive its corresponding evil–provided that this hatred is restrained within the limits of that which is lawful. These limits are:

1. Hatred must not be directed at the person of one’s neighbor; he is hated for his evil quality.

2. One may desire that the divine justice be accomplished in the sinner; but it must be a desire for divine justice, not a desire for the personal evil of another out of personal revenge.

3. The infliction of evil may not be desired absolutely, but only under the condition that the sinner remains obdurate and unrepentant.

4. It must be accompanied by that true supernatural charity which efficaciously desires the supreme good–the eternal happiness–of all men in general, not excluding any individual who is capable of attaining it. In a word, the sinner may lawfully be hated only when he is loved.

John L. McKenzie, “The Imprecations of the Psalter.” American Ecclesiastical Review 111 (1944): 91. Cited in John N. Day, Crying for Justice: What the Psalms Teach Us About Mercy and Vengeance in an Age of Terrorism.