The L1 Lagrange point itself is mildly unstable, but of course a beanstalk would be longer than the distance to the Lagrange point itself, and would extend into the Earth's gravity well slightly to keep it in tension. The beanstalk would be terrifically long, a minimum of 58,000 km just to get to the Lagrange point, and then depending on the counterweight mass, a bit further. However, because of the low lunar gravity, a Lunar beanstalk doesn't require an impossibly strong material to manufacture.
Information on Lagrange points can be found in the gravity reference pages at: http://www.merlyn.demon.co.uk/gravity.htm#L15.