That's a pretty old lamp holder and it's nearly indestructible, especially when operated by a wall switch. That being said it may not have been installed in a way that would be code compliant today. An electrician should be able to scope out the problem for you and either make repairs using what is already on the wall or in the wall, or giving you options like a flush mounted sconce that would be less likely to suffer damage from sticking out that far.

The problem could also be at the wall switch or elsewhere.

This fixture costs less than $25, has an integrated LED lamp and a shade, and sticks out just 3.75 inches from where it is mounted.

Jesco Lighting RE-GEO-FM-91011-3080-WH Relyence | Build.com

There is a tab in the center of the socket at the bottom, and sometimes those don't quite make contact with the bulb. It happens with other fixtures, like lamps, too. A quick way to check this is to make sure the power is off, get a small screwdriver, and use it to bend the metal tab up from the bottom of the socket just a little bit more. Then re-try the bulb. If that doesn't work, you can get an identical replacement for that porcelain socket for about $5-6 at your local hardware store. Just make sure the power is off before switching the wires to the new socket. And of course, before buying a replacement, you can turn off the power, undo the two screws, and pull the socket down to make sure there is no loose wire. Whatever is wrong, I can't see that it poses a hazard unless you decide to work on it without having cut the power to it.

Well elltwo and kudzu.....this is funny........DH came home from work and I told him about the light and what you all said, and the steps I had taken to try to figure it out. He said "Let me check something". He went downstairs to the basement and yelled "I fixed it." LOL........the last time he was in the basement he had left the switch down there that controls the light on the stairs switched only half way up.......so that's what caused the top light to not work. I never even realized we had a switch down there for the light at the top, since I only use the switch at the top. Well.......I'm glad we didn't pay an electrician alot of money to replace it all. haha Thanks again for your help!

It's good when the solution is simple, and even better that you didn't pay an electrician $150 to flip that switch for you. Plus, you learned something. :-)