In "Prometheus" why bring anyone but David-type androids?

An in-universe explanation would simply be that the people want to go there themselves. After all, Weyland wanted to travel himself when we could have simply sent a team of humans!

While a ship full of androids could probably complete tasks there'd be no one there making sure the tasks were done as they wished.

From a filmmaker's perspective a single android character creates a much more interesting story and set of interactions than humans and an android.

Without the guidance of a master, androids would ultimately fail on such a long and remote mission.

Despite David's wide range of abilities that he demonstrates through out the film, he fails repeatedly to make his own unique decisions, and when he does make his own decisions he makes poor ones.

• David monitors the dreaming of Elizabeth Shaw and invades her privacy.
• Ignoring commands to stop, David opens the chamber to the large human head which sets a chain of events into motion that ultimately gets the crew killed.
• David must report his progress to Peter Weyland and seek further instructions. Weyland orders him to "try harder".
• David infects Charlie Holloway and gets him killed.

David's are designed to do tasks that humans don't like to do. Not specifically tasks that they can not do. As mechanical servants they lack the free will to make their own decisions and as such are incapable of completing such a mission on their own.

At best, androids could travel to the planet, record data, conduct tests and report back. Much like a mechanical probe would do. You still need humans there who can ask questions like, "why did the engineers create humans?" or "why do they now want to kill us?".

That Answers Why They Didn't Send All Androids

The other reason is that Wayland was on board the Prometheus, and the only reason David was there was because Wayland trusted him. He thought of him as his son and did not trust his own daughter. The only people on the ship that knew Wayland was there were the security guards, David and Meredith Vickers.

Elizabeth Shaw thought they were on a mission to make a great discovery for mankind, but Wayland had other plans - to contact the engineers and convince them to extend his life. While Wayland could build androids that would never grow old, he himself could not overcome the fact of life - that we all will one day grow old and die.

One take as to why a scientific exploration wouldn't have just androids is the concept of whether androuds can own anything or not. In the world of science, usually the first person to discover something gets to name it. This can be taken as a sort of ownership. Can an inanimate object own something? Would the object's owner own what it discovers? It's a tricky philosophical concept when it comes to sapient non-organic objects.