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This is a much debated question and I am sure that there will be other answers for you to look at, especially in the discussion postings. I will take a different approach on this and think about the crucial element that is exploited by Frank R. Stockton to great effect: ambiguity.
Ambiguity is a quality that allows something to be interpreted in several different - sometimes conflicting - ways. Ambiguity adds complexity to a story. It can make fiction seem more like real life, where we often encounter people and events that are puzzling and mysterious. It is ambiguity that made this story such an instant hit and has kept people asking and pondering your question for more than a century.
Therefore, the whole point about this story is that there is no answer - the author is highly skillful in ensuring that there is enough evidence to suggest that the princess could open either door rather than just going for one - on the one hand you have her fierce devotion for her love, but on the other hand reference is made to the "hot-blooded semibarbaric princess." As Stockton himself says:
The more reflect upon the question, the harder it is to answer.
Part of the sheer genius of this story is that it keeps us pondering and guessing, never certain of what the princess will ever do.
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