Does Anemometer consumes power? I planning to add an anemometer to my small solar weather station.
I designed this system to run from a single 3.7v rechargeable battery and a solar panel so I have to calculate in more power drain.

What I don't really understand is do these anemometers consume any external power or just give out a voltage by themselves?

This one doesn't even have consumption specified:

https://www.adafruit.com/products/1733

For this Supply voltage : 5V - 24VDC:

http://www.ebay.com/itm/JL-FSX2-4-20MA-0-5V-Output-Wind-Speed-Sensor-Wind-Speed-Transmitter-Anemometer-/331521148973?hash=item4d3033482d
If it runs on electricity it definitely consumes power, its only how much the real question is. 

Since you are working with electronics I am going to assume that you have a multimeter, if you dont have one. Its about time you did, doesnt cost much and will save you a ton of time debugging. 

Nearly every multimeter has an onboard ammeter (the accuracy will be mentioned in the instructions manual). You can use this to determine the current consumed, have a look at a tutorial from sparkfun on how to do this