Are you asking if a single 12V wire, as typically found on a PC power supply, can handle 350W? The answer is no.
If you are asking what the maximum 12V power a particular 350W power supply can deliver, you need to provide the exact model. Or read the label on the side of the power supply. It will tell you the maximum supported wattage per voltage and the total power allowed (sounds like 350W)
If you want to know how to get the most out of a power supply of 350W @ 12V, then you need something that can draw 350W at 12V without melting. The power supply will likely hit its' over current protection or overheat running at maximum. Power supplies are intended to be oversized for their applications.