As a rule of thumb, I found there's less crime in Asia than in the Americas, and Western Europe. I'd be interested to know what horror stories you've read.
You have some violent crime in some parts of the Philippines, and plenty of (non-violent) scams in the touristy parts of Vietnam and Thailand (long distance busses). But with those notable and well known exceptions, I'd classify crime as rare.
Having been robbed a few times, I tend to leave valuables in the hotel/hostel now anyways, but I've found SEA to be safe all in all.