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I assume there exists at least an angle in which the 2 planes in which these 2 lines exist are parallel (else how do you define distance?).

Now draw a third line perpendicular to line 1. Measure the distance to when this line hits line2.

Does this make sense?

Though you can google the math formula for this problem, but I personally prefer to do it in this way:
for one of the two 3D line, express it in three linear equations:
x = a1 * t + b1
y = c1 * t + d1
z = e1 * t + f1
for the other line, you have coefficient a2,b2....f2

now the distance can be expressed as a function of t.
so a little simple math can find the min of distance.

by the way, the problem was part of this year's Google Code Jam Round1C prob2.

Nix: How do you guarantee that you can have two parallel planes always, why can't you have just two intersecting planes. Now how do you tackle the problem? check it out beibe !