Does Anemometer consumes power?
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