How do I make a rain detector on Minecraft?

You don't need one. There's this beautiful command called /weather clear. You can even specify for how long you want the weather to be clear. You can just run /weather clear 999999 each in-game day (employing a simple daylight sensor), and you will have no rain.

You can do more than that. You can disable the rain completely.

In Options -> Video you can disable rain particles (so it will rain, only you won't see it), AFAIK it is in Animations or Particles submenu.

Or you can use daylight sensor, because rain means drop in light -> weaker redstone signal, what you can make to start your redstone circuitry.

An active daylight sensor: powers adjacent redstone dust, including below it, and redstone comparators facing away from the daylight sensor, to a power level that depends on the time of day, the weather, and the daylight sensor's sky light level (see tables below) http://minecraft.gamepedia.com/Daylight_Detector

Rain means drop in light by 3.

And remember that you can rather use /weather clear [seconds] , so the nice weather will be long enough.

To test this out I created a machine that uses a snow golem to detect rain, to prevent lightning.

1. Snow golems take damage when killed, and drop items (Snowballs), so put it in the spawn chunk, and have a dropper under the hopper. The dropper should face into a chest.
2. Have a repeating clock ticking into the dropper, and also have a comparator coming out of the dropper.
3. Place a redstone dust next to the comparator and have a redstone torch on the other side of that dust then the comparator.
4. Then have a redstone repeater coming out of the output of the comparator, and set a redstone line with command block of your choice (weather clear, toggledownfall, say, tellraw @a ["Rain ceasing"], or watever you want).

There is a command in game that will turn off weather cycles, which means that it won't cycle trough rain,storm and clear anymore.

/gamerule doWeatherCycle true

If by any means the Cycle as turned off during any undesired condition, simply do

/weather clear

To clean it;

/weather rain

To make it rain!;

/weather thunder

To set it to storm.

It only works in version 1.11 and beyond, it works in pocket edition but does not work in bedrock as it was not added yet.

Source

You could also use a daylight detector. If it rains, the atmosphere gets a bit darker, so you can use ?-signalstrength for your advantage.