Well, first of all, lets remember that Eve was created out of Adam, so they both started out with exactly identical genes (except that Eve had 2 X chromosomes).
Also, I think we can accept the hypothesis that God created Adam (and therefore Eve) with perfect DNA. Perfect in the sense that there were no genes in either set of chromosomes that had a negative impact on the organism.
Now I also think we can assume that God expected them to breed and so he made the two sets of chromosomes as different as he could think of to instill a little variety at the start.
So in the first round of children (in fact all the children of Adam and Eve, assuming they remained monogomous throughout their long lives) the only variation between the children (and indeed their parents) resulted from genetic crossover and mutations. But with the assumption that the two sets of chromosomes possessed by A&E were as different as possible, this could result in some reasonable amount of variation between the children.
In the 930 years that Adam lived, he could have had a lot of kids. As they bred together mutations and crossover would actually result in an increase of genetic diversity, i.e. in the difference between the genes of the offspring and those of A&E. Of course, you also had first generation children of A&E (from their golden years) breeding with offspring up to 40-some generations removed from A&E. However, since we know that the original genes were perfect this would not reinforce flaws as inbreeding does today, but in fact restore positive traits that may have been lost in subsequent generations. Of course, the populations that had already migrated far from the land of A&E would feel less of this effect and would be free to genetically drift as much as they liked.
Obviously, the purity of genes that once allowed us to live just shy of a thousand years has long since disappeared, so this trait must have been dependent on having a very large share of the original chromosomal sets. As life expectancy dropped, generations became shorter and the process of genetic drift accelerated, allowing us to achieve the diversity we see today, a scant 7000 years later.
All part of the miracle that is life.