What's the best counter strategy against battlecruisers?

Unlike in Starcraft 1, the Starcraft 2 Battlecruiser is an over specialized unit. It has high ground damage and a long charge up time on its Yamato gun (excellent reference to the birth of anime right there). This combined with its slow speed give you a number of options to deal with it. I'll try to break them down by strategy rather than unit composition (as you should never build a specific unit to counter).

Zerg

The major Zerg strategies are as follows:

1. Melee Zerg - usually an effort to deal with a Mech heavy build, a late game switch to BC will punish this Zerg heavily. As Melee Zerg revolves around using infestors to pin down mobile units these are going to be your work horse. MC is nice, but Fungal Growth is going to be your key. Try to pin them down and swing your army around them. Their slow speed and tendency to bunch should mean you can take them out of the game.
2. Flying Zerg - Usually a muta harass strategy, you should see the BC long in advance or have Corruptors built to deal with air to air. However, if you don't for some reason you'll want to abuse the Battlecruiser's slow speed. Use hit and run tactics to string them out and then pick off members near the back. If you're having trouble with this strat, throw up some Spore Crawlers to slow them down. They won't do anything, but he'll take the time to kill them.
3. Missile Zerg - This is actually the hardest one (strangely enough) as Hydralisks die too quickly vs Battlecruisers. You're going to have to use creep and fungal growth to keep the range advantage. In a straight up fight you'll die so try to hit and run on creep. Nydus Canal help here.
4. Queens - its worth noting that mass queen is an amazing counter to Battlecruisers as the ability to heal and high air dps make queens tough and force BC to rely on Yamato to get any kills.

Protoss

The major Protoss strategies are as follows:

1. 4 gate - You should have High Templar by this time, if not default to the 3gate/Robo strategy. Templar feedback is useful (and you should use it), but the killer here is Psystorm. Battlecruisers not only move slowly, they have very slow acceleration. This means they'll be slow to respond to your storms. With 2-3 Templar you should be able to blanket them and throw down more as they run.
2. Starport - Any Protoss air unit does well against Battlecruisers. Phoenix shoot-and-scoot means you can stay out of range of running Battlecruisers, and their poor shooting method (rapid fire of low damage shots) means you will destroy them. Voidrays are actually less effective as you have to stay facing them and charge up. It isn't until you're fully charged that you'll be effective. If you can draw them out, this should be fine, otherwise sacrifice a building to charge them up.
3. 3 gate/robo - This is the hardest point to counter from as you'll only have Stalkers which get squashed by BC. Use your faster speed as best you can. If you can get blink, use blink micro to force them to spread their damage out. Don't blink in, this is a common mistake. Instead a-move in and blink out. Recharge your shields and attack as many times as possible.

Terran

1. 3 Rax - A MMM strategy will be able to produce cheap Marines which are more cost effective than any other unit. Even though they die faster, if you produce enough and constantly reinforce you should be fine. Use the 3 point strategy to maximize your marine's lifespan. Watch this quick video of Boxer. At the beginning you'll notice he spreads his marines out (here to reduce aoe from lurkers) he does this by taking a single control group and breaking it up into 3. This provide the ability to make a very strong "Arc." The result is he is able to bring more fire power to bay on the Lurkers than would otherwise be possible. Use the same idea against Battlecruisers. They'll be able to stack up for higher damage; forming an arc is your counter to this.
2. Mech - A Mech heavy build will naturally suffer against battle cruisers as your main defense is Thor which have little effect. Supplement your Siege tanks with Missile Turrets with upgrades. This will force him to kill the turrets while you remove your Siege. Your main force though, will depend on Thors with heavy SCV reinforcement to take them out. Where you can, add in Marines for high dps. Marines are the most cost effective dps you have.
3. 1:1:1 - This strategy is so effective against Battlecruisers that its most likely he won't build them, but if by some strange change he does use your Vikings. Vikings basically curb stomp Battlecruiser as they are everything the Battlecruiser is not: good vs air, long range, fast unit. Hit them at long range then run back. Since Battlecruiser shoots rapid fire weak shots, they should only get off 1 or 2, meaning you take minimal damage. If you lose this then you need to practice.

Remember, you should never think: "What unit counters X?" but rather "How do I use what I have to win?"

Basically, either a bunch of dedicated air-to-air units (vikings, corruptors) or mass marines/hydras/stalkers.

Race-specific tips:

Zerg

• Corruptors are really a hard-counter to battlecruisers, both because of their attack bonus (+ vs massive) and because of their special ability, which is particularly useful when fighting fewer but stronger units. Their high armor is also good against the battlecruiser's attack, which is composed of very high rate of fire but not that high damage.
• Mind control is great against any kind of strong unit, such as battlecruiser.

Protoss

• Void rays are useful against beefy units, and work pretty well against battlecruisers, as long as you have enough.
• Phoenixes are actually not that good, unless you get a lot. They die too quickly, and their relatively low damage means the high armor of the battlecruiser will stop a greater percentage.
• Feedback is pretty good against battlecruisers. Disables their yamato cannon and does a great deal of damage.
• Blink is really useful for sniping battlecruisers. Blink the stalkers right beneath them, then target just one at a time. This is why it's important to always accompany battlecruisers with some ground army, to prevent the enemy from getting underneath.

You shouldn't be giving him the time to get those 12+ Battlecruisers.

So you should press him earlier so he doesn't get that far up the tech tree AND has the time to built that many Battlecruisers