Can the players do anything to improve the crew quality? Or can it go down for any reason?

Yes and yes.

Reduced crew ratings could result from a lot of things - after prolonged battle with massive crew losses, pressganging new recruits might lower the crew quality until the new guys have learned the ropes while even stronger losses might permanently reduce the quality as the senior personnel of certain stations has bought the farm and can no longer train the newbies. Gaining better crew, on the other hand, would likely entail either training of some kind, replacement of unfit officers (though this may or may not result in morale problems) or recruiting from other, more well-trained starships.