The wires that go into and out of a converter (or power source, if that is what they call yours--it's the same thing) can be split into two categories: those that are related to the 120V power input side, and those that are related to the 12V power output. I think you're describing the positive 12V output wire as the bad one, but it's not perfectly clear. As BFL mentions, pictures would be helpful for understanding what you're seeing.
The 12V wires will have noticeably larger conductors than the 120V ones, as they carry nearly 10 times the current. The 120V wires may just be a standard power cable with a plug that plugs into an outlet, like any old appliance cord. Generally, there would be a minimum of three wires for the DC side for a standalone converter--one that isn't physically integrated into the power center chassis: the positive output, the negative output, and a chassis ground tie connection that's separate from the negative output even though both connect to the chassis ground at some point. Other wires would be for additional functions: some old converters had separate battery charging and 12V house power outputs, for instance, and some have controls or displays to show/select between charging modes, and some may have battery temperature sensors or remote battery voltage sense wires, and probably a bunch of other possibilities.
At any rate, wires get hot due to power being dissipated when current flows through them. When the heating is just right at a connection of some sort, it means that the connection wasn't solid enough and had excessive resistance, causing a lot of power to be dissipated right there and a lot of localized heating. If the whole wire is uniformly toasty, it means the wire was undersized or the current flowing through it was excessive (and the latter should be prevented by fuses or circuit breakers interrupting the current). For loose connections, the solution is to clean out any damaged wire and make a new, solid connection--of course doing so when all power is disconnected, including (for 12V circuits) the battery. Loose connections in RVs are not uncommon because the vibration and bouncing around of driving tends to cause screw terminals, etc. to work loose over time.