The Banjo Switch is a gray-white colored switch with Banjo's face on it. It is only seen in Banjo Tooie. You can only activate it by stepping on it as Banjo. You will need to use a Split-Up Pad to play as Banjo by himself. You can press A to switch to control of Kazooie after stepping on a Banjo Switch. These switches can be found in almost every world, and are usually next to a Kazooie Switch.
You will need to step on a Banjo Switch and a Kazooie Switch in the 9th and final world, Cauldron Keep. Stepping on both switches correctly will open the gate so you can progress to the 3rd Klungo fight.