We do use the term "kin" in BrE but rarely alone. "Friends and relatives" are sometimes referred to as "kith and kin" (Middle English origin, I understand).
The most common usage that I can think of is "next of kin" - when you are asked to provide emergency contact details by your employer or when undergoing surgery, you're normally asked to provide details of your "next of kin" - your closest family member so that they can be notified in case of any problems.
As various people have said, in most countries, first cousins are free to marry and procreate. It may not be a good idea for the reasons given, but it's not against the law in most places!
As far as the original question goes, I can't think of a way of saying it. Every website I've looked at simply describe the reasons for cousins not marrying as "the heightened risk of congenital defects" or a similar phrase.