The amount of light emitted by the solar corona is far too little to do damage. So during a total eclipse, but only during totality, it is completely safe to look at the corona with naked eyes. NASA is a good source of information about eclipses, including safe viewing and photography techniques. According to the NASA Eye Safety page:
The Sun can be viewed safely with the naked eye only during the few brief seconds or minutes of a total solar eclipse. Partial eclipses, annular eclipses, and the partial phases of total eclipses are never safe to watch without taking special precautions. Even when 99% of the Sun's surface is obscured during the partial phases of a total eclipse, the remaining photospheric crescent is intensely bright and cannot be viewed safely without eye protection [Chou, 1981; Marsh, 1982]. Do not attempt to observe the partial or annular phases of any eclipse with the naked eye. Failure to use appropriate filtration may result in permanent eye damage or blindness!
So it would be a good idea, especially if the eclipse is not near local sunrise or sunset, to be aware of how long totality is expected to last, and be ready to look away as soon as you glimpse the "diamond ring" effect where the disk of the sun first becomes visible through a notch on the moon's limb.
Photography of the eclipse with a DSLR requires that you look through the camera. If the mirror were locked up, then the camera would be projecting a nearly in focus image of the sun onto the shutter curtain, and eventually onto the film or sensor. Clearly, significant damage to your eye and/or the camera is possible. To avoid damage, a solar filter should be used in front of the lens. NASA also has a good resource page related to eclipse photography.
During the 2006 eclipse I was very close to the center of the path of totality on a yacht in the Mediterranean, attempting photography with a hand-held DSLR with a 500mm lens. I had a Thousand Oaks Optical black polymer solar filter on the front of the lens, and it was fun to see the disk of the sun with decent magnification. However with that filter in place, the whole frame was dead black when pointed anywhere in the sky that didn't include the sun. Aiming was a bear. While practicing, I made the mistake of opening my other eye to check the larger view which was not a pleasant experience.
For use aboard ship, I punched one eye out of a cardboard pair of eclipse watching glasses that happened to be made with the same filter material I had on the lens. With the camera on the unfiltered eye, and the other eye filtered I was able to successfully aim and photograph the partial phase without further pain or injury.
During totality, I removed the filter and shot the corona with the same lens.