The circuit is divided into two halves, there a "hot" side and the secondary side. It would be next to impossible to blow up anything on the hot side by attaching the batteries backwards.
You can use an ohmmeter to measure the capacitors. Measure with the leads in one polarity, then reverse the leads and watch the meter closely. When you first make contact, there should be a reading that rises rapidly and goes back to an 'open' reading. If you read steady resistance, that would be bad. If you don't see any breif readings upon switching polarity, that may or may not be bad. A small (less than 10uf) capacitor may be so fast your meter won't catch it. It depends on your meter too.
You can also measure across the legs on IC102 and the diode on the output side. Reverse the leads. Look for something that reads shorted in both directions.
If it blew the large chip on the bottom, you're probably toast.