Actually, Drunken Sailor is Aeolian, not Dorian. It follows a chord progression of i minor to bVII or v minor. It's more convenient to play in in E minor on a D whistle. Here's the reason. Eminor would be the i minor chord and the flat seventh would be D major. The regular or (natural) seventh of the e minor scale would be D#. You would flatten the D# to D to get the flatted seventh. So the song is built on the Aeolian scale of E minor which would be E,F#,G,A,B,C,D,E.
The C# doesn't appear in the melody so it means that the song is Aeolian. The chord progression for the Dorian mode would have to emphasize the C# therefore an A major chord or an F# minor chord would have to be in the song.

Frank Hamilton