C Rags answer is the best one so far because it is based on Scripture (Unfortunately as of now I can't comment neither vote up on his answer).
If I am to give another answer, I would also use the Scripture beginning in Romans 5:12-14 (KJV):
12 Wherefore, as by one man sin entered into the world, and death by
sin; and so death passed upon all men, for that all have sinned: 13
(For until the law sin was in the world: but sin is not imputed when
there is no law. 14 Nevertheless death reigned from Adam to Moses,
even over them that had not sinned after the similitude of Adam's
transgression, who is the figure of him that was to come.
This text (and its continuation) establishes Adam as the head of the natural race, and "him who was to come" (=Jesus) as the head of the spiritual race. It also establishes that each one has an effect over all the ones born in their respective race. In the case of Adam, sin is one such thing; death is the other one.
There are several genealogies in the Bible that establish Enoch as being a descendant of Adam. As such, he inherited sin and death from Adam.
Verse 14 of the quoted text says death reigned from Adam to Moses, as to establish this concept of race headship, meaning people were dying not because they disobeyed God in the same fashion as Adam did: "that had not sinned after the similitude of Adam's transgression", or that there was a law for them to trespass and be condemned of death, but they were dying simply because they were born into the race of Adam.
The very text of Genesis 5 begins saying that Adam was first created in the image/likeness of God. Then we know he fell and became a sinner because he ate the fruit of the wrong tree, then Genesis 5.3 says he "begat a son in his own likeness, after his image" (no longer in God's image, but in his sinful image):
1 This is the book of the generations of Adam. In the day that God
created man, in the likeness of God made he him; 2 Male and female
created he them; and blessed them, and called their name Adam, in the
day when they were created. 3 And Adam lived an hundred and thirty
years, and begat a son in his own likeness, after his image; and
called his name Seth.
Enoch, being listed as a descendant of Adam, was no different that Cain and the others who died, in the sense of having inherited the sin problem and the death problem from Adam. Is was only Jesus, who was born from a virgin Miriam, that did not appear as a human descendant of Adam (recall from verse 3 only the male passes on the sin and death nature). Instead, he's the first one of the spiritual race, and heads up the spiritual race. The others follow suit by means of a spiritual union with Christ in crucifixion, burial and resurrection, that occurs when they born again.
Now one can cite Romans 5:14 and argue that Enoch was the only one between Adam and Moses that did not die. Genesis 5 explains why he didn't die: it was not because he wasn't born with a sin and death problem, the text plainly states he was a descendant of Adam, but because (as wax eagle properly put) God imputed Christ's righteousness on him and then took him up.