Polling people after the fact is tainted heavily.
Polling BEFORE anything happens is also tainted. What if there's a hereditary component? There have been 'crib deaths' in my family (going back 3 generations, to grandparents siblings). If someone has a relative who has died of SIDS, of course its going to be more on their minds.
What about people who had close friends or family members who had a serious medical problem with an infant, again it would easily raise the paranoia levels (which c'mon, the worries are all there anyway).
Toss in issues with first time parents. I think it's a fair statement to say that most parents I know worried much more about their first child, on average. If they worried more about a child that was not the first, it was due to medical problems or special issues with that child. That would point to very real concerns.
And not only that, let's toss ALL this stuff out. There are alarms you can use on an infant that you believe has a higher than average chance of SIDS (if another sibling has had SIDS for example). If I were to have a child, and I had some type of omen or vision or some strong feeling of dread where I really believed that would happen, I'd get it. I wouldn't care what assurances others gave me, I'd get it for peace of mind if nothing else. Or I'd argue that perhaps somehow I'm picking up on a piece of information I'm not able to verbalize, but it's scaring me. Anyway even if its costly, it wouldn't be cost prohibitive when you weigh the factors.
They should do a poll about pregnant women and how many have nightmares that something terrible has happened to their child. I'm a woman, I don't even HAVE kids and I've had nightmares like that.
I'm more sorry than words could express that people lost their babies like that, but that 'study' just isn't close to being sound. :\