There are plenty of well-intentioned evangelicals who have a hard time with the obvious meaning of Genesis chapter six – for a variety of reasons. The problem is that, like denying the existence of the devil altogether, denying this particular satanic attack makes it more difficult to understand the reasons why God created mankind in the first place. The entire five part Satanic Rebellion series lays all of this out in detail, including the passage in question (see the link: "The Nephilim" in part 5). Perhaps the most important biblical teachings that will necessarily be misunderstood from a failure to grasp the truth of what happened in Genesis six regards the nature of antichrist who is himself the off-spring of the devil (see the link: "The Origin of Antichrist"); he is the serpent's seed (i.e., "your seed", anti-Christ, who opposes "her Seed", the Christ). Failure to understand this will make those who think such things "impossible" very vulnerable to the beast's claim that he is truly the Messiah, "the Son of God" -- how else, the argument will run, could he possibly do such miraculous things?
Here are my translations of the pertinent New Testament passages which discuss this event:
It was also by means of the Spirit that [Christ] visited the [angelic] spirits in prison (i.e., in the Abyss), and proclaimed [His victory]. [These are the angels who] were disobedient in the days of Noah at the time when God patiently waited (i.e., delayed judgment) while the ark was being built.
1st Peter 3:19-20a
For if God did not spare angels when they sinned, but plunged them into Tartarus with its gloomy pits (i.e., the Abyss), preserving them for the [day of] judgment, and did not spare the antediluvian world, but kept safe Noah as a proclaimer of righteousness and the seven with him when He brought the flood upon the ungodly inhabitants of the world, and condemned the cities of Sodom and Gomorrah to destruction, reducing them to ashes and making them an example to those bent on similar ungodly behavior, and rescued righteous Lot who was tormented by the depraved lifestyle of those lawless men - for through the things he saw and heard just by dwelling among them this righteous man was damaging his righteous way of life day by day on account of their lawless deeds. For the Lord knows how to rescue the godly from temptation, and to keep the unrighteous under punishment for the day of judgment - especially those who in their lust pursue the polluting of the flesh and so despise [God's] divine authority.
2nd Peter 2:4-10a
Although you know all these things, I want to remind you that though the Lord saved [all] His people from the land of Egypt at the first, in the end He destroyed those who proved unfaithful, and the angels who did not keep to their own realm but deserted their proper habitation He has imprisoned with everlasting chains in the gloom below (i.e., in the Abyss) in anticipation of the judgment of that great day, just as He did with Sodom and Gomorrah and their neighboring cities, for they all prostituted themselves in the same manner as these [angels] did, having pursued sexual relations (lit. "flesh") that were inappropriate for them (i.e., outside of the natural order). [And so it is that] they have been appointed an example, and incurred the penalty of eternal fire.
All three of these passages clearly teach angelic involvement of procreative kind in the run up to the flood. In the light of the above, it is very difficult to maintain that Genesis six is not talking about fallen angels; if these passages are not "literal", then what do they mean? Would they then mean anything?
The Nephilim were on the earth in those days and afterwards as well (i.e., both before and after the start of the 120 grace period before the flood). For when the sons of God went in to the daughters of men, they bore to them those "mighty-ones" (i.e., the Nephilim) whose names are famous from ancient times.
There is no reasonable explanation for the fact for how these Nephilim could be in any sense "mighty ones" apart from their unique parentage, just as this passage says: they are unique because they are the offspring of "the sons of God" rather than "men"; as the passage above distinguishes between those two class, any contrary interpretation would have to show just who the "sons of God" are since they are not "men".
As to the "sons of God", in fact, this phrase beney ha-'elohiym ("sons of God" in the Hebrew) and its variants most often refers not to men but to angels (Job 1:6; 2:1; 38:7 [anarthrous]; Ps.29:1 [cf. Ps.103:20; 148:2]; and Ps.89:6: beney-'elim). I'm not sure what "proof" this person requires. I would think that in reading the first two chapters of Job it would be obvious to anyone in Job 1:6 and Job 2:1 these "sons of God" are angels. Have a look. Whom else could they be there? Some things are beyond the necessity of serious argument. Here are more links to places where I discuss these matters in greater detail:
The Nephilim in Genesis 6
"The Nephilim" (in SR 5)
The Origin and Fate of the "Giants" of Genesis Chapter Six.