If 1st cousins have a child there is a higher risk of genetic diseases being passed on to their child (source below). One of the best ways to prevent this is to ban marriages between 1st cousins, in the same way people are already banned from marrying certain other relatives.
You may be interested in these active petitions
Registry offices should be given the power to order independent DNA evidence if there are suspicions a couple are 1st cousins.
Source: https://www.genetics.edu.au/publications-and-resources/facts-sheets/fact-sheet-18-when-parents-are-relatives-consanguinity