I've done a bit more research on barrage balloons and about how they could form a better deterrent to aircraft.
Balloon aprons could be deployed by having each balloon anchored to a separate ship (or a single large ship, with one balloon anchored to the bow and the other to the stern), thereby providing a kind of mobile shield that could be deployed where needed around a battleship. This could be used to defend against side, rear or frontal attacks. The cables are there to ensnare aircraft, but I’m also thinking of a variety that could have explosives contained in the cables that could be detonated on contact with flying anti-ship torpedoes or aircraft. Some historical barrage balloons carried explosive charges that would be pulled up and come into contact with aircraft, making them a sort of aerial mine.
You could possibly have a similar design for warding off attacks from above (or maybe even from below?), with the cables forming a web that would be hung horizontally between the balloons. I’d imagine that the presence of balloons above a ship would obstruct dive-bombers, and carpet bombing would have to take place at a higher altitude, forcing bombers into the range of concentrated AAA fire (WW2 AAA mounts didn't have the fast-tracking ability of modern AA guns, which made defending against fast low flying aircraft more difficult). As a result of this, bombing accuracy would be decreased. If a captain chose to hover above a city or civilian infrastructure then this would probably deter any bombing run, assuming that the enemy doesn’t want to inflict civilian casualties.
You could also perhaps have anti-ship mines, such as larger versions of the explosive-carrying balloons, or maybe some kind of Flugstein-powered device that has some sort of low-tech homing ability.
To counter the balloons, they could be simply shot down or, as the Luftwaffe attempted in WW2, have aircraft outfitted with special cable cutting devices attached to their wings. I can imagine that the presence of well placed balloons (combined with effective AAA and a fighter screen) would make downing a ship more of a challenge. In gameplay terms it would be fun trying to dodge and/or shoot down these obstacles, so as to pave the way for a bomber strike.
I partly got the idea of including such things from the old Carrier Command game; those of you who may remember it might recall those expendable, programmable decoys that you could deploy around the carrier in a particular formation to distract incoming missiles. Although I've tried to tweak the concept of barrage balloons in order to apply it to this setting, I think that I might not have thought it through well enough. You can probably come up with reasons why barrage balloons wouldn't be needed to be applied to aerial fleets, so feel free to pick apart these ideas.
Some sources of info that I've been looking at:
http://en.wikipedia.org/wiki/Barrage_balloon (a general summary)
http://www.airpower.maxwell.af.mil/airchronicles/apj/apj89/sum89/hillson.html (an interesting essay-like article which covers their history and suitability for modern-day air defence - well, as modern as it was in 1989)
http://www.worldwar-two.net/armamento/90/ (interesting article which covers their effectiveness during WW2)
http://longstreet.typepad.com/thesciencebookstore/2008/04/the-semi-encaps.html (plans for balloon deployment around London leading up to the war)
http://www.skylighters.org/barrageballoons/ (more history and some pictures).
Edit: Replaced bad link in first para with a working one, added an extra link and altered a few sentences.