Is government funded space exploration a waste of money?

Maybe. The Chinese are right to send a core drill to the moon. We need to do the same on Mars. We've already been fostering a sub culture that lives in subterranean mines in places like Alaska. If we can establish underground environments on Mars, we have a start.
NASA has been finding out the "law of the land" regarding our adaptation to space like no private venture possibly could. Of course, the ISS should be expanded at the fastest possible rate, so we can manufacture materials that are better suited for space habitations. There are manufacturing methods that can only be done in space, like the production of metal "foam", where the void spaces that would be bubbles are instead vacuum. You can do things with metals like magnesium, titanium and aluminum in the void of space that can't be done with our use of inert gases on earth. Why aren't we rushing to form an international space agency?

 

I don't think there is a magic budget number. If our presence does not add to the body of knowledge I would rather see the money we spend on it better spent on projects related to space that do add to our body of knowledge.

 

It is entirely feasible that by 2063, human technology could be to the point that Gene Roddenberry's prediction could come true. However, before the Great Bird of the Galaxy's predictions, we already have a technology from Star Wars. The ion engine, two of which powered the famous TIE fighters and gave them their name are already used by human spacecraft.

I don't think it should go that far. I'd like to see them have around a seventh of the military's budget. Use it to develop an entirely reusable horizontal-launch SSTO with enough payload capacity to surpass the Delta IV Heavy. Once one of those vehicles is in service, a truly permanent space station could be built on orbit that could serve as a multiuse research station, on-orbit spacecraft construction yard, and a spacedock for future exploration.

 

Provided we don't kill each other off, in time the planet will kill us, so figuring out how to get off this rock before it kills us seems like good sense...