I went tumbling down our very steep stairs when I was about 7 with my mum and the cat. Have a tiny tiny scar on my lip that can only be seen if I pull my lip up and point it out.
My mum felt awful about it for ages, but for me it was just a cool story to tell, especially the bit where I came into school at break time and my best friend stopped everyone from hounding me.
Dp has a very long very nasty scar that runs the length of his entire forearm that he acquired by falling a small plastic kids chair and breaking his arm very badly. Again, just an accident that could have happened to anyone.
Dc also slipped off a chair when put for lunch with dp and has a scar on his face where he was caught by his glasses, but that too was just one of those things. It was a slippery chair. Nothing would have happened had he not been wearing glasses.
And me, when dc was a few months old, before I thought he could roll over (having never showed any signs of it) rolled right off the changing table when I went to fetch another nappy. That felt pretty awful but he was fine ten minutes later. Even the doctors in the hospital (hit his head so got him checked out) said that it happens to the best of us (and told me their tales of unfortunate accidents with their own child)
Like pp have said, guilt does seem to come with the territory but we really all have these stories and it doesn't make you a bad parent.
It's a massive learning curve and accidents happen to us all.