It doesn't work for me, and I doubt it will work for you. There is an extension for Banshee to accomplish this, but it normally causes banshee to hang, sometimes crash.
If you are using 1.9 from a ppa, install the community-extensions, and then enable stream recorder. When you go to the radio browser, you can then click on the record button and see if it works for you. If it doesn't work I would suggest using rhythmbox to record.
I don't know whether this is the best way, but I created a play-list of Armenian radio stations, Armenian.m3u
#EXTM3U
#EXTINF:-1,Beirut, Lebanon - Bashdee
http://yp.shoutcast.com/sbin/tunein-station.pls?id=546514
#EXTINF:-1,Istanbul, Turkey - Nor Radyo
http://www.norradyo.com:8000/
#EXTINF:-1,Moscow, Russia - Armenia.ru
http://radio.armenia.ru/pls/listen.pls
#EXTINF:-1,International - Armenian Bible Study Radio
http://sc7.mystreamserver.com:8036
#EXTINF:-1,International - Armenian Pulse Radio
http://yp.shoutcast.com/sbin/tunein-station.pls?id=416989
#EXTINF:-1,International - Arm Radio
http://www.realradios.com/x/stream/50536
#EXTINF:-1,Yerevan, Armenia - Public Radio of Armenia
http://www.armradio.am/eng/live/listens.pls
#EXTINF:-1,Yerevan, Armenia - Public Radio of Armenia - Shoutcast
http://yp.shoutcast.com/sbin/tunein-station.pls?id=898669
#EXTINF:-1,Yerevan, Yerevan, Armenia - AR Radio Intercontinental
http://www.realradios.com/x/stream/50541
#EXTINF:-1,Yerevan, Yerevan, Armenia - Yerevan Nights Radio
http://www.realradios.com/x/stream/46211
#EXTINF:-1,Boston, MA - Hairenik Radio
http://www.live365.com/play/acaa
#EXTINF:-1,International - AIM Radio
http://www.live365.com/play/aimmedia
So I can maintain this list separately and then import it into Banshee after upgrades.
I wish, however, for Banshee to be more user-friendly with add/remove/update/export/import of on-line radio stations, as this way seems to be not very intuitive (at least to me).
Best Answer
I think this is not a quite useful answer for you, I'll rather show you or some nice person with time and ambition a way to go.
Because I didn't find an existing possibilty to get what you want (and I think there really is none), I believe the easiest way would be to write a tiny tiny extension. And this really doesn't seem very hard because the code exists already in other extensions.
For example Radio Station Fetcher: it just searches for radio stations on some websites and adds them to your radio "playlist". Therefore it uses a single dialog with two buttons (one for genre search and one for free text search) and that's all (you can try it yourself by installing
banshee-extension-radiostationfetcher; you can find it then in the "Tools" menu).I looked at the code and all one have to do is replace the code for searching stations on those website with code that reads your list of stations. All the other code can almost stay the same. I would do it, but I don't have time.
You can find the code of all extensions here. Live Radio has the same functionality of adding a list of stations to your radio "playlist" but I think it has a bigger codebase to look through.
I hope that will help you or some kind person who will do the work for you :P