Before the introduction of a vaccine in 1963 measles caused about 4 million cases annually in the USA, leading to 48,000 hospitalizations and 500 deaths. That’s a lot of kids in the hospital and a significant mortality. The measles vaccine has nearly eliminated these things.  
         Today even with a 90+% vaccination rate a measles outbreak costs $12,000 per case in lost work time for parents, hospital treatment and tracking.  
         In 2013 WHO estimates 145,000 deaths world wide from measles. In the US we are isolated from the deaths because of our health care system but it doesn't mean that skipping the vaccine because a few people die from it is a good choice. Common complications are ear infections (5% to 9%), lung infections (1% to 5%), convulsions, and, in 1 out of 1 000 cases, inflammation of the brain (encephalitis) that often causes permanent brain damage or mental retardation. Blindness, loss of hearing and long term decreased in lung function are all also possible.  
       I don't agree with mindlessly sticking needles into our children but to refuse without knowing the fact is plain stupid. If you don't travel outside the USA, keep your children away from places where they can be exposed to the disease and our vaccination rate stays above 90% (95% ideally) they are pretty safe. Not a risk I would take with my child though, if truly worried I might wait until they were 18 months old to vaccinate.  
      Its easy to forget how susceptible to infectious diseases humans were before vaccines. Small pox killed in England alone 23,000 people the year before they started vaccinations. 1967 The year WHO made small pox vaccinations a priority 14-16 million people were infected and 2 million died. The last recorded case of small pox was 1977. Pretty convincing argument for at the very least some vaccinations.