It'd have to be one of Kirk's - there's the ever famous "Risk is our business," of course, but I'm even more fond of his address to the crew after Balok scares the bejezus out of them in "The Corbomite Maneuver."
Those of you who have served for long on this vessel have encountered alien life-forms. You know the greatest danger facing us is ourselves - an irrational fear of the unknown.
But there's no such thing as the unknown - only things temporarily hidden, temporarily not understood. In most cases we have found that intelligence capable of a civilization is capable of understanding peaceful gestures. Surely a life-form advanced enough for space travel is advanced enough to eventually understand our motives.
All decks stand by.
It works because he's not speechifying so much as just talking - focused but almost conversational in tone - and it's not all bombast and oratorical generalities, but reassures people and calls them back to their duty in a crisis by reminding them of who they are and what they know to be true.