Take a story's temperature by studying its tone. Is it hopeful? Cynical? Snarky? Playful?
Whimsical, Magical, Dreamy, Hopeful
The play's opening scene is set in the midst of a murderous storm, which would seem to suggest that The Tempest is going to be as dark as, say, Macbeth, which also happens to begin with a terrible storm.
This play, though, is most definitely not Macbeth. Soon enough, we learn from the tricksy spirit Ariel that not a soul on the ship was harmed. (Seriously, the play has a "tricksy spirit." How can you argue with the tone being "whimsical"?)
Thus, even naturally scary events are undone by the magic of the island, and the tone that seeps into the rest of the play is one of wonder, amazement, and admiration. Mystery still abounds, but the magic performed is not black and scary, rather more a thing of the natural world. Further, with its silly drunkards the play has a certain lightness to it—even the would-be killers of the King tell hilarious jokes and are occasionally lighthearted.
In The Tempest, all past wrongs are forgiven and even the nastiest people get second chances. We think Miranda's final speech pretty much sums up the play's hopeful attitude about life and the human condition:
How many goodly creatures are there here!
How beauteous mankind is! O, brave new world
That has such people in 't! (5.1.215-218)
Sure. Miranda may be a little naive but we like to think this is Shakespeare's way of telling us that life is pretty marvelous.