I generally call the tool I use for this "a sound guy". You can automate it though by doing some dynamics processing on the input to the effects unit. You need to use a device called an expander on the feed going to the effects unit (or have an effects unit that provides it's own expander).
An expander is a lot like a compressor, but the ratio goes in the other direction. Up until a certain threshold, the audio is normal, but then after the threshold is reached it goes up more quickly. Thus, the input level to effects would be low until the threshold is reached and then would rapidly increase after the threshold is reached.
You could do the same thing with a gate if you simply wanted it on and off, though that might result in some odd effects unless the gate can fade on and off. If you do go the gate route, you probably want a slow release time.
The end result is that the signal level in to the processor is low or off when there isn't much input but elevated/on when the threshold is exceeded. This can then be mixed back in at a static wet/dry level and you get a changing overall wetness since the input to the effects processing is increasing more than the dry level.
