Is there a genetic basis for having dry eyes? Is it familial or hereditary?

Not really. There are some dry eye diseases associated with sjogren's syndrome, which may have a genetic and environmental roles, but its probably not dominant or recessive as we know it. Familal and hereditary are terms sometimes used interchangeably, so I'm not sure what you're asking. Most dry eyes are prob not genetic based.
Usually not. Dry eye is not usually genetic although disease that can cause dry eye can be genetic in nature. Most dry eye is not from inherited problems though.