I don't believe it matters during a jump if the donor vehicle is running or not. There are nothing but old wives tales to support this claim and no technical foundation.
Reversing polarity will usually, bag the rectifier, and overload green and red wires between the rectifier and the battery. There is no fusing in this circuit.
If the rectifier shorts instead of opens during the failure, then any power connection will melt wire insulation.
Check/measure the six rectifier diodes, and replace as necessary. Also, replace any wires found with damaged insulation.