The Book of the Watchers (1 Enoch 1-36; pre-200BC)
The book of Jude makes the relevance of 1 Enoch for the NT beyond question. Jude 14 quotes the first part of 1 Enoch, called the Book of the Watchers: "Enoch, the seventh from Adam, has also prophesied about these individuals, 'Behold, the Lord comes with ten thousand saints to effect judgment on all and to convict every soul in relation to all their deeds of ungodliness that they did and all the harsh things ungodly sinners have spoken against him" (quoting 1 Enoch 1:9). It seems reasonable to conclude that Jude considered some portion of 1 Enoch actually to come from Enoch and, perhaps, even to be Scripture.
Since Jude already quotes the Book of the Watchers, it is easy to hear Jude 6 in the light of 1 Enoch as well. Jude 6 says, "the angels who did not keep their beginning but left their first dwelling have been kept for the judgment of the great day in eternal chains under gloom" (2 Pet. 2:4 incorporates the same material). While Christians naturally understand this statement in the light of Satan's fall, it seems more likely that Jude has in mind the fall of the watchers in the days of Noah. The sons of God (angels) have sex with the daughters of men (human women) and giants are born of the union. But God has the angels Raphael and Michael bind these angels in the valleys of the earth (1 Enoch 10:4, 11-12) until the day of judgment.
1 Enoch 6-11 tells the story as it was commonly understood by many Jews and early Christians. Tertullian in fact believed that this story held the key to 1 Corinthians 11 and the veiling of women. Women praying and prophesying, interacting with the spiritual realm, should have a veil on their heads because of angels like the ones that slept with women before the Flood. I don't personally go with this interpretation of 1 Cor. 11, but it does shed some light on how the ancients understood angels.
First, the ancients conceptualized angels as males, including male organs. There are no female angel names in the Bible or ancient Jewish literature. Secondly, not all angels are good. Paul mentions that Christians will judge angels (1 Cor. 6:3). It is possible that he has these same angels in mind! The question at least arises whether Jesus' comment that all will be like the angels in resurrection pictures the resurrection body as angelic (cf. Thomas 104).
It also seems most likely that 1 Peter 3:19-20 has this story in view. Christ, "having been put to death in the flesh but made alive in spirit, in which also having gone he preached to the spirits in prison who disobeyed formerly when the patience of God waited in the days of Noah..." What other Jewish tradition involves the binding of spirits who sinned in the days of Noah? The idea is that as part of the end of the age, the resurrected Christ proclaimed judgment to the spirits of the fallen angels.
There are some more general ways in which the Book of the Watchers might give background to the thought of the New Testament and Jesus' ministry. One of the hallmarks of Jesus' ministry was his casting out of demons. How is it that demons have come to rule the earth? For 1 Enoch 12-16 (not for 1 Enoch 10:15), demons emerged from the dead bodies of the giants. God tells Enoch to tell the fallen watchers: "you originally existed as spirits, living forever... The spirits of heaven, in heaven is their dwelling; But now the giants who were begotten by the spirits and flesh--they will call them evil spirits on the earth, for their dwelling will be on the earth. The spirits that have gone forth from the body of their flesh are evil spirits" (15:6-9--Nickelsburg's new translation).
Not of immediate importance to the New Testament, but important background as we consider Jewish understandings of the afterlife, is 1 Enoch 22. In this passage we find four compartments for the dead. Those in two compartments are set. These are criminals who were punished on earth and the righteous who apparently died peaceful deaths. Those who face unfinished business are the righteous who were unjustly killed and the wicked who apparently died in peace. This chapter reminds us that not all Jews looked to a general resurrection. Indeed, it is not clear that we should think of a bodily resurrection in this chapter at all.
Astronomical Book (72-82; ca. 200BC)
I can't think of any significant background that this portion of 1 Enoch provides for the NT.
The Apocalypse of Weeks (93:1-10; 91:11-17; ca. 200-170BC)
The NT does not directly interact with this historical apocalypse. It is significant generally in the fact that it does not consider the current temple legitimate yet has no sense of a coming messiah or resurrection. There is perhaps a sense that the fallen angels will be replaced after they are judged with new "powers of heaven." Not clear at all from this context that these are resurrected righteous.
The Dream Visions and Animal Apocalypse (83-90, esp. 85-90; ca. 160BC)
Again, no specific interaction with the NT. Now there does seem to be a somewhat messianic figure along with the new temple (90:37), but in my opinion there is still no resurrection. The fact that "all the sheep" are in that house is more likely a reference to the gathering of Israel from the Diaspora (90:29). I am not convinced that the abyss full of fire (90:26-27) is different from the Valley ge Hinnom, since bones burn.
The Epistle of Enoch (92-105; some parts 1st century BC)
An extensive part of the middle section of this part of 1 Enoch is not attested at Qumran, leading Gabriele Boccaccini to suggest it was written after the group went to the Dead Sea (ca. 100BC). One of the striking passages in this section for me is 103:4: "The souls of the pious who have died will come to life, and they will rejoice and be glad; and their spirits will not perish." This is not a bodily resurrection. It is some sort of strange resurrection of spirits, perhaps like 1 Enoch 22.
Where this datum potentially impacts our understanding of the NT is realizing that there may have existed a category of resurrection among the Enoch-Essene trajectory that was not physical resurrection--as in 2 Maccabees and perhaps the Pharisees--but spirit resurrection, whatever that might mean.
And here let me give my support for this translation of Acts 23:8: "For the Sadducees say that there is no resurrection, neither angelic or spirit resurrection, but the Pharisees confess both." Whether this datum sheds light on Paul's spiritual body or 1 Peter's, "having been made alive in spirit," I don't know. It does, however, correct the (I believe) misconception that the Sadducees did not believe in angels. That idea is based solely on a faulty translation of this verse.
Similitudes of Enoch (37-71; late 1st century BC)
Perhaps the most important background 1 Enoch provides for the NT comes from the Parables of Enoch. This section of the collection is not attested at Qumran and the mention of the Parthians may date it to around 40-30BC. The most important intersections with the NT are:
1. The "Son of Man" as the Messiah. Jesus of course regularly referred to himself as the Son of Man. But what did he mean by this phrase? It was probably as ambiguous to his audiences as it has proved difficult for scholars to agree on. At times Jesus seems to use it to say little more than he is a guy, a son of man (Foxes have holes... the son of man has nowhere...). At other times the phrase has all the significance of Daniel 7:13, such as when Jesus tells the high priest that he will see the Son of Man coming on the clouds of heaven (Mark 14:62).
2. The Son of Man seems pre-existent in some way. The Parables at times came into focus in the early 80's in the question of how Christians came to see Christ as pre-existent. The dating of the Parables was a significant factor in that equation. The Son of Man was named before the stars of heaven were made (48:3). This "light of the nations" (cf. the servant of Isaiah 42) "was chosen and hidden in his presence before the world was created and forever" (48:6). It is possible, however, that the Son of Man is portrayed here as Wisdom (cf. 42:1).
3. The third parable features the Son of Man seated on a throne of glory judging the angels and all humanity (61:8; 62:2, 5). Earlier it is said that Sheol would give up all its dead, our first sense of general resurrection in the Enoch corpus (51:1 in the second parable, which also involves the Chosen One sitting on a throne, 51:3).
The parallels of these images with Matthew in particular are strong. Matthew 25:31-46 speaks of the Son of Man sitting on a throne with his angels with him, as in the Parables. Matthew in general has more fire, weeping and wailing, etc. as we find in 1 Enoch. This imagery is not nearly so prevalent in other intertestamental literature. Further, Matthew seems to adjust various Q material to portray Jesus as divine wisdom.
Of course the idea of Jesus on a throne judging the world also appears in Revelation 20:11. And Jesus destroys the lawless one with the breath of his mouth in 2 Thessalonians 2:8, just as the Chosen One slays all the sinners with the word of his mouth in 1 Enoch 62:2.
All in all, it seems very likely that Jesus and the authors of the NT were aware of and influenced by the Enoch corpus. It of course goes beyond the evidence to suggest that Christianity might have grown significantly from the Essene tradition. But that doesn't stop me from wondering...