I already know that crossing over causes genetic variation. My question is that since DNA is a stable molecule, why would it undergo process like crossing over during which it becomes so unstable and develop a chance of mutation?
Crossing over is believed to be beneficial for evolution. It does not cause only mutations, it allows for the switching and mixing of genetic information. Thanks to crossing over entire protein families evolved, like the hemoglobins. Indeed it is one of the mechanisms that shaped the chromosomes. Some in-silico simulation proved that, in general, adaptation take place faster if crossing over is applied.
Here some extra references for further readings.