Up until the 20th century, cousin marriages were very common (most people lived in small farming communities, where ALL the eligible guys / girls were your cousins).
I forget where I read about it, but the risk of bad recessive genes coming up is not much higher, and it is counterbalanced by the fact that the couple will be more likely to share values (resulting in a stronger marriage).
To be specific, children of cousin couples are two times more likely to have birth defects, but this increase correspond to an increase in probability from 3% to 6% for an individual child to have birth defects. Roughly, from 1 in 33 in non-related couples to 1 in 17 in cousin couples.
The real risk comes from multiple generations of cousin marriages, because that decreases genetic diversity and makes cousins closer to siblings. That's at least 25% or 1 in 4 children with birth defects.
Also, multiple generations of cousin marriages causes
inbreeding depression, which is a fancy way to say children are becoming prone to getting sick and quite literally losing IQ.