No, it was never particularly safe, not since it started growing in the 1960's. The surrounding area has always had criminals preying on tourists.
Disney used to be immune because they kept most of their stuff inside the theme parks, which are very secure, with rapid-responding security forces. They're frighteningly fast...even a verbal altercation will be efficiently surrounded in seconds by large men with ear-radios, the people involved whisked out of sight before most "guests" in the area even notice. And I'm quite sure that any group that looks "suspicious" has manned cameras following their every move for the entire duration of their visit. That, plus the fact that few gangsta sorts are likely to pay $60 for a park admission!
But the moment they opened a larger shopping/club district open to the outside world, they got the real world's troubles. Way back in the early 90's, when I was there as a kid, when Downtown Disney was still Pleasure Island/Disney Marketplace, there was a gang stabbing at an arcade, and numerous fights.
The parks are pretty locked down and watched carefully. But the shopping district is a high-traffic tourist area with lots of criminals and illegals, just like any such place in the country.
