Hard to say. I expect there'd be some of that, but then again, the more time you have in front of you, the more you have to lose by doing something grossly stupid. Not that most teenagers think that way, but accident rates tend to decline sharply once you reach adulthood, and stay flat until you hit old age - I don't have the statistic handy, but I believe that the death rate due to accident/injury in those 75 and older is the highest of any age group, even dumb kids, likely because age makes you more frail. Perpetual youth would eliminate that, so basically, if you were smart/lucky enough to survive to adulthood, you'd probably have pretty good odds of living for a very long time. Anyway, your odds of dying violently are lower than they were a hundred years ago, which were in turn lower than a thousand years ago, so I don't think it's too unreasonable to expect the world to continue to get somewhat safer in the future.
Who knows, though? Maybe after hanging around for a thousand years, life would come to be kind of boring, encouraging more thrill-seeking among the millennial set ;)
I donno. If we knew we were going to live more or less forever, without getting sick, there might be those who would, say, decide to spend the next 100 years as heroin addicts. It's only 100 years, so why not? It's not a choice I'd make, but for others, with nothing to lose ... it's hard to know how immortality would affect things.