Aliens Steal Cable: Used, but not thoroughly. Even though the aliens had been studying the Earth for a while, Klaatu had apparently never heard Bach before.
Aliens Are Bastards: The aliens are committing genocide against a sentient race to protect a planet which is in no appreciable danger. Even if we wipe ourselves out life will go on, and CO2 and methane scrubbing bacteria will come in and clean up after us. This is instead of just giving us better technology. They had to get to our level of technology, to get to theirs.
Humans Are Bastards: It's been estimated that about 3 species become extinct every day because of our actions. Humanity is just one species. So if we're going to go out anyway, they don't want us taking any more species with us.
Artistic License - Biology: "His life-support suit was similar to a placenta." "That makes sense, because a placenta sustains life."
Broken Aesop: We're destroying the other species on our planet, and aliens think that's bad. Fine. So why does Helen's love for Jacob change Klaatu's mind? A mother's love for a child of her own species, while charming, doesn't really show anything except a desire to perpetuate her species. It'd be more valid if she showed love for an animal, perhaps something completely dissimilar to humans. Instead, The Power of Love conquers all.
Grey Goo: How GORT was going to wipe out humanity.
Humanity on Trial: Klaatu had authority to judge Earth. In the original, Klaatu's mission parameters exclusively covered delivering the message. The message being that humanity could either join the galactic civilization or stay on Earth. If we stayed on Earth, then we would be left alone. But if we decided to join, then we would be expected to obey their rules, and be annihilated unless we gave up our warlike ways.
Klaatu Barada Nikto: The line is spoken once, by Klaatu himself to stop his robot's defenses from destroying everything when he is shot by a nervous soldier. However he was in alien form at the time and the line is so garbled and distorted you really have to listen closely to discern the famous words.
The military, law enforcement, and the government in general, though all of government in all of fiction is guilty of this trope, and it's not an unexpected reaction to aliens being suddenly real. Klaatu comes to Earth and reaches out his hand to the protagonists. Clearly, putting a bullet in him is the appropriate response. Only later do they realize he was able to shut down their defense network on a whim, and so they decide imprisoning and (implicitly) torturing him is a good idea. Klaatu's decision, after consulting a spy on Earth, is naturally that Humans Are the Real Monsters and have to go, thus the swarm of nanobots begins devouring every man-made object in its path. The military bombs it, only to see it grow larger. The Secretary of Defense at least grows a brain at this point, but the president orders even more bombing as if the opinion of his military adviser isn't worth considering.
Klatuu's people deserve extra stupidity points as well. Consider that their entire motivation for destroying humanity is to preserve the non-human portions of Earth's biosphere. They then set their nanotech-based weapon on "Dissolve Everything", including rocks and trees! In the immortal words of Robert Asprin, "Very inferior as superior beings go."