Shortwave broadcasting might not seem like much these days, but recent events in the Middle East have shown pretty well how vulnerable modern means of communication like cellphone networks and the interwebs are in times of crisis. It sure didn't take much to get the whole of Egypt offline.
Even satellite radio/TV is vulnerable. Jamming shortwave broadcasts isn't quite as easy.
Now suppose you want to get information into a country that's cut off otherwise - what do you do? Borrow the DoD's flying 4-transmitter FM station? Might work for a few hours or days, but would certainly get a bit expensive in the long run. Set up a MW transmitter ship? That takes a while to arrive. Buying some airtime at a shortwave transmitter site, by contrast, is quite feasible.
Since you bring up the BBC, the renowned World Service had to face severe budget cuts early this year since, well, Britain is kinda broke. And it's not like they hadn't operated on a shoestring budget for years. So even the classic MW frequency broadcasting to Europe had to go, a step met with regret by a number of listeners on the continent. That transmitter had been operating since the days of WWII. The closure of services in Cantonese to China was greeted by Chinese officials in a triumphant press release - China has invested quite a bit into shortwave in the last decade, which is pretty hard to miss on the bands.
There are a number of people who consider the cuts a short-sighted move. Absolutely speaking, international broadcasting still is a pretty inexpensive way of making your voice heard in the world and promote democracy, human rights and freedom of information. From what I remember, Germany's international broadcaster Deutsche Welle (admittedly, rarely the one to deliver programming that would knock you off your socks) had an annual budget of like 1..2 million a few years ago, not even 1/1000th of a % of total national spendings. I would bet a lot more money just "disappears" in questionable projects. (We've had our fair share of airports noone uses and bridges to nowhere.)
Shortwave may be less interesting over in the US (religious broadcasters and kooks... ugh), but over here it's definitely good for a number of stations from areas that normally aren't on your radar. And with about a hundred bucks shipped from China (or, in fact, Amazon in the US) for a perfectly usable portable receiver from Degen or Tecsun, there is little reason to complain about associated cost. (If you don't want to go with something ready-made, making an antenna is a good opportunity for learning a whole lot, too! And if you ever want to get on the air, be prepared to learn a lot more still.)
