Burnerman, good evening.
If the meter box is recessed into the wall what is probably happening is that either, as markstev27 has posted previously rain water is entering at the top and to a lesser extent the sides of the box then running to the rear of the box and causing the problem.
What can at times, but mostly in very cold weather is that because there is no insulation, or outer brick skin, having been removed to insert the recessed box holding the meter Etc. the cold external air has in effect direct access to the cavity side of the inner leaf of brick or block that your property is probably constructed from, the lack of insulation on the inner surface of the recessed box may be causing the problem?
it is simple enough to Re-Mastic the joint between the recessed box and the external surface of the recessed box, but, to try to get any Insulation on to, the rear surface of the recessed box, and the inner surface of the recessed box ait easy, after all the box in effect belongs to the utility Company who will probably not allow any tampering at all.
No doubt the board will have some Ideas on the above?
Ken