The best tool to use is a digital multi-meter. You can probably get one for 10 to 20 dollars.
The connections between the cartridge and the RCA plugs must be continuous. There are several points where there might be a problem, such as the internal wire right behind the RCA plugs getting broken by the previous owner shoving the receiver against the wall, and bending them too often.
The meter allows you to positively identify the problem area, without having to guess.
Headshell - RCA plugs
Red wire - Right center
Green - right skirt
White - left center
Blue - left skirt
The contacts between the headshell and the tonearm might need cleaning. A shot of contact/control cleaner, and the eraser on a pencil, will serve nicely.
The clips on the headshell wires might need a shot of the same, and a bit of movement.
It is also possible to check the cartridge. the resistance across the pins of each channel should read several hundred Ohms, closely matched.
Keep them spinning.