You should consider the materials you will be joining, are they carbon steel, stainless steel or some alloy like chromemoly.
Also consider the thermal characteristics of the operating environment. Is it a high temp application? Looks like the 316L is designed to withstand corrosive stresses at high temps.
That 316L has a high nickel content and the top hits on a google search say it is good for joining molybdenum bearing materials - I assume you want to use the 316L with a stainless steel.
If you simply want to transfer torque, the quality and geometry of the weld is the most important. Cracks in the weld surface or defects in the weld will result in accelerated fatigue damage. So choose the process that minimizes the chance for cracks and defects. Also, if you will be doing the welding, PRACTICE. There's no substitute for experience in welding.
I have had a great experience using ER70S2 on chromemoly with the TIG process. The extra Si in the electrode helps to deoxidize and seems to bring contaminants to the weld surface. I imagine the MIG process will produce similar results with such a material combination.