You can go at this from two angles: Do the speakers sound like they should, and are the components in the crossover doing what they should?
As to the first, if everything looks OK (drivers not trashed, etc.) and you can put your ear up to each driver and hear music, step back and see if they sound balanced or whether something is wrong.
As to the board, about all you can do (short of having an oscilloscope and audio frequency generator) is check individual components - resistors, capacitors and inductors. Usually you'd have to unsolder one end of each to check them. If everything looks OK and the drivers all work, rather than tearing the xovers apart, most people will replace capacitors based on age.
If you think something specific is wrong but there is no visual evidence on the board, you could burn a test tone CD and play various frequencies to see if there is a problem somewhere.