Theoretically you can just as easily present ID to register to vote in other forms, such as a birth certificate.
However, most of these “Voter ID” laws make the point of requiring a *photo* ID such as a drivers’ license to be presented at the polling station. Which a birth certificate is not.
People lose or misplace their drivers’ licenses all the time. I did the other day, and it took me a few days to get into the DMV, a couple hours out of my day and about $25 to get a replacement.
Trivial investments of time and money to me but not all are as fortunate.
More information on how state laws differ about this: http://www.ncsl.org/research/elections-and-campaigns/voter-id.aspx