We don't feel unsafe at Disney and while most accidents there are the fault of guests who ignore rules and throw caution to the wind, not all are.
Notwithstanding the recent example of the Epcot bus crash (where seven people were injured due to no fault of their own whatsoever), let's look at some history:
July 29, 2005: 25 riders are injured when a coaster train on California Screamin at the California Adventure Park rear ends another train. Subsequent state investigation concludes accident was the result of a faulty brake installed by Disney.
September 5, 2003: a male passenger is killed and eleven others are injured on Big Thunder Mountain at
Disneyland when a train derails into a rear-end crash. Subsequent state investigation concludes the accident was the result of both improper maintenance and improper employee training.
Christmas Eve, 1998: a male guest is killed and his wife is seriously injured when a cleat from the
Columbia sailing ship at Disneyland tears loose when sujected to the stress of a taut line, shearing across a group of waiting passengers. Subsequent state investigation concludes improper employee training was at fault. Disney eventually settles a lawsuit from the survivors of the deceased guest for a figure experts believe approached $25 million.
January 3, 1984: a woman riding the Matterhorn Bobsleds is thrown from her car and killed when struck by another ride vehicle. Subsequent state investigation shows her seat belt had come unbuckled,
but was not able to determine whether it malfunctioned or was accidently/deliberately opened.