I couldn't have said it any better than Trieste and Pumpkin Seeds.
Signing a petition is a public declaration, and your opponents have the right to know who you are, if only to verify that you exist. To my mind, if your ideals aren't strong enough to overcome potential downsides, maybe they are not important enough to sign a petition on in the first place.
Now, it is possible that some people will attempt to use this disclosure to harm their opponents, which they certainly do not have a right to do. However, as Brandon points out, any such attempt is going to be illegal and action can be taken in response to that in civil or criminal court depending on the nature of the case.
And this does not prevent anonymous letter writing campaigns, contributions, etc. There are plenty of ways to support a cause anonymously. But a petition should be a list of names, individuals, instead of just a raw number. It gives the process a sense of legitimacy, accountability, and weight.