My Independence Institute colleague Rob Natelson examines the question from an originalist perspective, in a new working paper published on SSRN. His analysis is summarized in this blog post on his website. In brief: political contributions are best analyzed as a form of Freedom of the Press. The Freedom of the Press includes the right to anonymous authorship. The right can be breached in cases of abuse, as when a civil libel plaintiff needs to discover the identity of the person who libeled him.
Some readers may disagree with the first part of Rob’s analysis, but the point about the right to exercise the Freedom of the Press anonymously seems indisputably correct. Rob extends the anonymity argument far beyond the points made by Justice Thomas in his Citizens United concurrence.