"Turn on the microphone" is the most common way to express the action of opening the microphone in Engpsh.

来源：英语的说法来源于直接表达开启麦克风的动作，使用动词"turn on"和名词"microphone"。

Examples:

Can you turn on the microphone so we can start the conference call?
Please turn on your microphone so I can hear you better.
I need to turn on the microphone in order to record the podcast.

翻译：

请打开麦克风，我们就可以开始电话会议了。
请打开您的麦克风，这样我就能听得更清楚了。
我需要打开麦克风来录制语音节目。