I get the spinning circle when my music app is open, it never turns off! Can this drain my battery? on brand new iPhone 5s on ios7
Apps can turn on and off the indicator at will, so it's not a 100% reliable indicator that data is actually being transferred.
You could make an experiment: leave the phone idle for two hours and check how much energy is consumed with the app on and off, respectively.
As for battery drain, YES it will drain the battery, but it is not the spinning circle that does it, it is the Network data transfer going on (in your case playing music).
Apple in its Wisdom and Generosity added that information for your to know there is a network activity going on.
The WiFi signal only says you are connected, but says nothing about are there data been transferred.
So now you have the additional information telling you when data transfer is happening that:
could help you managing your data plan (if you pay per use or have some sort of limit in your data plan ect.)
tell you your system might be updating some apps