First of all, the Statue of Liberty is not constructed of bronze. It is an iron frame with a skin of copper sheeting. This New York Times article says that before around 1902 the statue was a dull copper color, but that patination began to spread then, fully covering the sculpture by 1906. As /u/art_con states, patination actually protects the metal, unlike rust, which continues to eat away at iron. That green color is good - if it were removed, it would come back again. Eiffel and Bartholdi undoubtedly would have been aware of the patination of outdoor copper sculpture, and would have not only planned or this effect, but indeed counted on it.
This painting from 1886 depicts the unveiling of the Statue in that same year. I suppose it is as close as we will be able to see to the "original" color of the copper sheeting, since color photography would not come into widespread use for much longer.
Here also is a Currier and Ives print of the sculpture, though any color used by that company should be taken with a whole shaker of salt. They were interested more in inexpensive and fast production than slavish devotion to realism.
This NPS site has a trade card that might also show what the statue looked like, though, like I said, there is no way for us to know "for sure" what she looked like in 1886.
The best book that I know of about the Statue is Darcy Grimaldo Grigsby's Colossal: Engineering Modernity - Suez Canal, Statue of Liberty, Eiffel Tower, and Panama Canal, if you're interested in a scholarly account of the Statue that deftly and powerfully places it into a transatlantic culture of technological modernity and imperialism.
I can't answer your question, but I think it's relevant to your question to note that outdoor bronze sculptures are often patinated; meaning the formation of a passivating oxidation layer is induced through a chemical process (passivation refers to the fact that once formed, the chemical oxidation process stops and no further metal ions undergo oxidation).
Just to further clarify your question: rust is a term that is typically used to refer specifically to the oxidation of iron containing metals (iron oxidizes in a rather unique way in that the oxidation of iron typically produces an increase in volume and the oxidation continues to eat into the metal, which is why it is particularly destructive and why there is a specific unique term for iron oxidation).
