How does one respond to the charge that the New Testament endorses superstition by its references to demon possession?

Demon possession was a real, historical phenomenon of the first century. Spirit entities, known as demons [KJV "devils"] did inhabit and afflict human bodies during that age.

Scripture dos not spell out the question of demon origin, though Bible scholars have proposed several theories. Some may be readily dismissed, including the theory that demons were the disembodied spirits of a pre-Adamic race of men who lived in an alleged "gap period" between Genesis 1:1 and 1:2. No evidence of any such gap exists and Adam is called the "first man" (1 Corinthians 15:45). Others speculate that demons resulted from the cohabitation of angels with antediluvian women (based upon a misunderstanding of Genesis 6:1-4), but Christ taught that angels are sexless beings, incapable of such unions (cf. Matthew 22:30).

Two more plausible views surmise that: (a) demons were the spirits of wicked dead men whom God permitted to leave the Hadean realm to indwell some people;1 or, (b) demons were fallen angels who were allowed to escape their confinement (Jude 6) for a similar purpose.2 Regardless of origin, the New Testament recognizes the fact of first-century demoniacs.

As to the nature, demons were spirits (Matthew 8:16) -- not physical (Luke 24:39). As to character, they were spiritually unclean, evil and under Satan's immediate control (Matthew 12:24, 43, 45). Demons were intelligent (Mark 1:24) and could exercise both volition and locomotion whenever permitted (Matthew 12:44-45). Demon possession frequently brought about physical and/or mental illness (distinguished from the demon itself, cf. Matthew 4:24). Dumbness (Matthew 9:32), blindness (Matthew 12:32), convulsions (Mark 9:18), epilepsy (Matthew 9:32) and supernatural strength (Mark 5:4; Acts 19:16) were characteristic of demoniacs. No reason is given as to why demons entered particular individuals; they inhabited men (Matthew 9:32), women (Luke 8:2) and children (Mark 7:30).

Apparently, demon possession was temporarily permitted by God in order that the supreme authority of Christ might be shown. As the Lord revealed his control over nature (Mark 4:30), disease (Mark 1:12), material things (John 2:9) and even death (John 11:44), even so, he also demonstrated his power over the spirit realm (Luke 11:20). Jesus' authority over evil spirits amazed his contemporaries. They exclaimed: ". . . what new doctrine is this? for with authority commandeth he even the unclean spirits, and they do obey him" (Mark 1:27). His disciples, by his authority, could also expel demons (Luke 10:17), except on one occasion when hindered by their weak faith (Mark 9:28; Matthew 17:20).

With the termination of the supernatural era of the early church, demon possession, and the corresponding gift of expulsion, ceased. Satan's supernatural power was bound (Matthew 12:29). Certainly, the devil exerts great influence today. However, as God does not work miraculously in this age, but influences men through his Word and providence, so also, the devil wields his power indirectly and non-miraculously through various media. Certain "modern" cases of supposed demon possession are doubtless the results of psychosomatic problems, hysteria, self-induced hypnosis, delusion and such like. They have natural, though perhaps not always understood, causes.

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1. Alexander Campbell, "Demonology," Popular Lectures and Addresses.
2. Charles Hodge, Systematic Theology.