Just read this piece from Wired Science on this editorial in Public Library of Science Biology. It seems that marriages between first cousins remain illegal in over 30 states of the USA. So, if they were alive today and American (an odd thought!), the marriage of Emma Wedgwood and Charles Darwin would be banned in most of America (Charles' mother, Susannah was sister to Emma's father, Josiah Wedgwood II).
Charles did worry about the potential genetic ill effects of marrying his cousin. However, the editorial in PLSB reports that the risk of ill effects in the offspring of first cousin marriages is only marginally greater (~3%) than in the general population and is about equivalent to that seen in mothers over forty. Given that no one has ever suggested that laws should be brought in to prevent births in the over-forties, the authors of the PLSB piece argue that laws banning first-cousin marriages are unjustified.
Among the children of Charles and Emma, three died in childhood (Mary Eleanor Darwin at less than four weeks old; Annie at age ten and Charles Waring at six months). I am no sure whether there is any evidence to suggest a cause of death for Mary Eleanor. Randal Keynes has suggested that Charles Waring, born when Emma was 48, suffered from Down's syndrome. However, the baby's death from scarlet fever may or may not be related to his mother's age or parents' cosanguinity. I am not aware of any increased susceptibility to scarlet fever in Down's syndrome, so it could just be a coincidence that the baby suffered from both conditions. The cause of Annie's death is unclear. Tuberculosis has been suggested but the issue has never been subjected to scrutiny in the peer-reviewed literature.