Like others have explained already, your birth certificate is one of the few documents that is accepted as official identification which does not include a photo.
Normally, a single non-photo id alone is not sufficient, for obvious reasons. As said above, you typically need to combine two or more non-photo identification documents for acceptance, like your social security card, utility bills, medicare card, gun permit, etc. It varies from state to state.
Why do we keep saying “state to state”? Because the assumption is that you (or the bad guy who stole your birth certificate) is attempting to get some form of government issued photo id… like a drivers license, which are issued at the state level. In Texas, for example, the non-photo identification requirements to get a drivers license are listed here.
Once you have a drivers license, then getting a U.S. Passport is a piece of cake, because you only need that with the birth certificate and you’re good to go. A passport implies a lot of legitimacy.
But in order to really conduct commerce (i.e. steal an identity to gain access to that person’s credit), you must have their social security number. But, if you’ve got a drivers license and birth certificate, it’s easy to get a new social security card. Here’s how.
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The problem with the multiple forms of non-photo id approach in use today is that it’s basically a variant of security through obfuscation. That is, that it’s difficult to gather the various pieces need to successfully assume someone else’s identity. As we all know, it’s actually not all that hard.
So… are you (or your identity) in danger with your birth certificate out in the wild? I would say maybe, but probably not. Like @jp mentioned, it’s easy to get a copy of a birth certificate already. To me, it would depend on why it was stolen. Was it nabbed with a bunch of other valuable stuff, like jewels from a safe or something where you think they were really trying to get the other goods? Or do you think someone really wants to hijack your identity? Unfortunately, when it comes to criminal behavior with one’s own goods, it’s likely best to prepare for the worst and hope for the best.
To be safe, you might consider getting the security freeze on your credit, like @Npt suggests. Again, the process varies from state to state. Try starting here.