I have a 36" RCA which I purchased soon after they came out on the market(about 15 years ago). Lately, I've been having a problem where the picture is suddenly condensed into about a 6" band that runs horizontally across the middle of the screen. The sound stays on. What is perplexing me is sometimes, it can be remedied by tapping or banging on the side of the t.v. Sometimes this works. Sometimes this doesn't. Should I be looking for a new TV or is this fixable?

Sounds like bad/cracked solder connection, you will have to look carefully at circuit board.
Remove solder and resolder, don't overheat. Look for verticle output, or any of the big devices. Can be hairline crack. Can be small part also. Start with the big stuff.