The reason that solar and lunar eclipses are about equally common, is that they occur when the moon is "close enough" to the plane of the ecliptic at the point where it is full (lunar eclipse) or new (solar eclipse). Otherwise it completely misses the sun (solar eclipse) or the earth's shadow (lunar eclipse), and you don't get even a partial eclipse.
If the moon orbited precisely in the ecliptic plane, then there would be a total lunar eclipse every full moon, and every new moon a solar eclipse that's either total or annular according to the distance of the moon from the earth at that moment. But the moon doesn't orbit in that plane, it intersects it twice a month, at a time that most months isn't when it's close enough to being lined up with the earth and sun.
"Close enough" in the case of a solar eclipse is when the earth, as viewed from somewhere on the sun, is partly obscured by the moon (therefore: part of the sun is obscured by the moon when viewed from part of the earth). Whereas "close enough" for a lunar eclipse is when the moon is partly obscured by the earth from the somewhere on the sun (and therefore part of the earth's shadow falls on part of the moon). This is just about the same size target area.
What the sun "sees" as the moon orbits the earth is similar to what we see when we look at the moons of Jupiter: the moon moves from one side to the other and back. Sometimes it passes "over" the earth, sometimes "under", sometimes through. What's required for a partial eclipse of either kind is that it moves through, such that those two disks intersect as viewed from the sun. The difference from Jupiter, is that Jupiter is really big, so passing through is more likely for the moons of Jupiter than our moon.
If you really want to know which kind is a few percentage points more or less frequent, then you're beyond my ability to predict. Consult a table of eclipses covering a long period of time, or wait for an answer from someone with better qualifications :-)
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