This is a follow-up post to let everyone know how this all turned out.
Although my above post wasn't completely clear, what I had done at that point in time was use silicone caulk on the top, the entry points on the top, around the top "cover" of the meter box, and across the top and down along the sides everywhere that the meter box met the wall of the house.
After I did the above, and after the next rain, I still had water coming in. So, that didn't work.
I still wasn't able to arrange for the electric company to come out and meet me there so they could open the box and I could see what the problem may be. Instead, they just showed up unannounced one day when I wasn't there, looked at the box, didn't do anything, and told the people who live there that the caulking that I had done is all that they would have done and, supposedly said I did a good job.
and,
After reading what Mr. E wrote, and having given up on the electric company for any help in opening the meter box, I decided to do even more caulking of the meter box. I used more clear 100% silicone caulk and caulked all around the meter itself, and vertically down along the sides where the front cover of the meter box meets the box. I also decided to caulk all around the round mystery lock on the front that only the electric company can open. So, now the meter box is completed sealed shut with my 100% clear silicone caulk. I don't know if the electric company would think that was a good idea or a bad one, but that's what I did.
Fortunately, that worked!
We have had tons and tons of heavy rain for the past two weeks and no water is coming into the electric panel in the basement -- it is completely dry. That means that the water leak problem was coming from the meter box itself with water getting in there somewhere. The masthead (or whatever it is called) above apparently was not a source of the problem since caulking the meter box did the trick.
Here are two photos -- a before and after shot:
Unfortunately, the silicone caulk is clear, so it is hard to see, and I had to take the photos with my cell phone which doesn't take very good pictures. But, if you look at the second shot carefully, you may be able to see that I caulked everything -- the top, the sides, around the meter, around the round lock on the front, etc. And, that solved the problem.
Thanks everyone for your help.