The SID chip is used almost solely for producing sound, and is not required for a C64 to boot. Thus a faulty SID predominantly causes audio-only issues without otherwise affecting the functionality of the computer. They often fail partially, producing generally good audio but with for example some notes skipped or low volume. Sometimes the symptoms are more prominent, like a whole channel missing, or no audio at all. Although rare, a bad SID CAN prevent boot-up, or cause crashes as it heats up. The chip is almost always socketed, so easy to rule out via removal. Some software uses SID for generating random numbers, and the chip also controls paddles. So if those don't work properly but everything else seems fine, suspect the SID.
CAUTION: When swapping SID chips, make sure you don't mistake a PLA for a SID. They are the same size, and the chip locations vary depending on the board revision. You risk killing either chip if you put it in the wrong socket and turn on the computer. The SID has better chances of surviving this mishap, but the 12 volts in the SID socket will quickly kill the PLA.