I've indeed heard and read this argument since I became interested in space.
But are we really honest with ourselves when we say so?
I have increasing doubts.

What does the public want? Humans rather than robots? Why?

It's because humans are associated with TV coverages, dynamic pictures, sharp movies...In a word : the public wants good images !

If unmanned robot missions provided us with as sharp and varied and dynamic pictures, the public would get a lot more interested!!
We've seen that trend with the robots on Mars. If an unmanned spaceflight towards Mars could show us a live coverage from the rocket of the launch, orbital phase, Mars Insertion phase then while cruising, pictures of our dynamic sun (H-Alpha filter), telescopic views of our receding planet, of the nearing Mars, of the stars, asteroids, comets, of itself from a few meters away, later a "live" coverage of the landing with the approaching landing site, the actual touchdown, then a wheeled vehicle leaving the platform while that rover would take a camera that once in a while could be left on a spot so that it could take movies of the rover moving or digging or climbing...then I'm sure the public will not only enjoy all those various images coming in all the time, but will also identify with the robot, giving it an affectuous name...
And those robots could be directed by our best scientists on Earth!
All this for a tiny fraction of what a manned spaceflight to the same place would cost.

Movements, fast events, slower events, all caught by an army of artificial eyes, never getting us bored thanks to the never ending imagination of their operators. Who knows, one of those cameras could even have the privilege of returning back to Earth, be exposed to the public in a museum, known to the world so that anybody who'd want to touch something that went there, could do it instead of wondering where such and such astronaut is (if he is available to the public and still alive and not too expensive to approach).

Why is the public interested in humans going to space? Because it wants to know how weak and "human" they could be, faced with a difficult situation? But no space organism will ever send humans to space if it's not almost certain nothing too dangerous will happen to them. Robots are a totally different matter.
Here, it's about exploration, discovery, finding out, not about migration or colonies. We can't discuss the latter before the road has been cleared by the former.

Raoul