The Characters' Opinions:
Dr. Scott clearly believes that the mission was not intended to benefit humanity. When Frank'n'Furter uses the Sonic Transducer to immobilize Brad, Janet, Scott, and Rocky, this exchange takes place:
DR. SCOTT: You won't find earth people quite the easy mark you imagine. This sonic transducer - it is, I suppose, some kind of audio-vibratory, physiomolecular transport device....
BRAD: You mean...
DR. SCOTT: Yes Brad. It's something we ourselves have been working on for quite some time. But it seems our friend here has found a way of perfecting it. A device which is capable of breaking down solid matter and then projecting it through space, and - who knows - perhaps even time itself!
JANET: You mean he's gonna send us to another planet?
FRANK: Planet Schmanet, Janet!
Eddie clearly agreed. In the dinner scene, when Scott is singing Eddie's Teddy, he says that Eddie warned him in a note which reads (What's it say, what's it say?):
I'm out of my hed.
O hurry or I may be dead!
They mustn't carry out their evil deeds!
Eddie had inside access to Frank's schemes, more so than any earthling aside from Columbia, so if he considered Frank's plans evil, we should probably take his word for it.
The Possible Missions:
The Sex and Kinkiness:
This was clearly not the mission; it is the reason the mission failed.
Frank N. Furter, it's all over.
Your mission is a failure;
your lifestyle's too extreme.
Frank's lifestyle was only extreme in the sense that he was a hedonist; if hedonism was the mission, it would have been a rousing success.
This was also not the mission - if it had been, the mission would have been considered a success, and Riff-Raff certainly wouldn't have killed Rocky. Frank made Rocky simply because Eddie wasn't attractive enough for him.
FRANK: Oh, I just love success!
RIFF-RAFF: He's a credit to your genius, Master.
MAGENTA: A triumph of your will.
COLUMBIA: He's okay!
FRANK: Okay? Okay? I think we can do better than that! Well, Brad and Janet, what do you think of him?
JANET: Well, I don't like men with too many muscles.
FRANK: I didn't make him for you! He carries the Charles Atlas seal of approval!
And after killing his previous sexual plaything Eddie, Frank consoles the traumatized Rocky, explaining why he created Rocky in the process:
FRANK: Oh, don't be upset. It was a mercy killing. He had a certain naive charm, but no muscle.
Later, Riff-Raff confirms that Rocky is merely a diversion, not part of Frank's mission:
Your new playmate is loose and somewhere in the castle grounds.
This explains why Riff-Raff was so hostile to Rocky, literally from the moment he was "born":
The Real Mission:
The Sonic Transducer:
Dr. Scott may have been onto something when he suggested that perfecting the Sonic Transducer was Frank's main objective, but his assessment of what it was intended to be used for is probably wrong. After all, the Transit Beam Riff-Raff mentions when he apprehends Frank projects matter through space, and it was presumably how Frank, Magenta, and Riff-Raff got to earth in the first place, so there was no need to "perfect" it. The Transit Beam and the Sonic Transducer must be two separate devices.
It is important to note that the Transducer has no effect on Magenta, Riff-Raff, and Frank. It only affects Scott, Brad, Janet, Rocky, and Columbia. This suggests that the Transducer only works on earthlings, and is harmless to Transsexuals. Thus, if one wished to build and/or perfect the Transducer, one would need to test it on earthlings. The best way to do that? By going to earth.
What does it do? It bends humans to the will of Transsexual Transylvanians.