Why do the machines keep humans alive if they have nuclear reactors?

Well, whatever the reason machines keep humans alive, it is definitely NOT because of humans' excellence in the role of batteries. However it is possible that the machines themselves are spreading that myth. (For example Morpheus would believe anything the Oracle says). That machines don't really need humans for power is revealed in the last part, when the Architect agrees to free all humans from the Matrix.

The actual reasons may be much more complicated, and much more humane. The machines just don't want humans to become extinct, just like we humans are trying to protect Siberian tigers, as well as other endangered species. Even though a tiger may kill a human given the chance.

However protecting humans is much harder than the tigers. You can't put them in a cage, because then their minds will degrade. You can't let them roam free, because they probably won't survive in the new world, or because they will probably keep trying to fight the machines until they kill themselves. Humans aren't very rational, you see.

As a solution, the tandem Matrix - Zion was created. The Matrix is conveniently situated right next to the Zion so that people in Zion can focus the majority of their energy in "freeing" others from the Matrix. Also, quite conveniently, the Matrix lacks any real defense mechanisms.

If there was just Zion without the Matrix, the humans would be trying to develop nuclear weapons or EMP bombs in order to launch the assault on the machines. Obviously they would fail, but they will also kill themselves in the attempt.

If there was just Matrix without Zion, then such system would run a risk of collapsing and destroying the humankind if some unforeseen catastrophe occurs. The machines aren't really good at adapting to sudden changes. And they are vulnerable to other unpredictable factors (such as Agent Smith). The machines know that, but they can't do much about it.

Now why do they need to destroy Zion every now and then? Well, first of all they probably don't see it as much of a big deal -- the population of Zion is comparatively low. Plus they want to keep them from acquiring too dangerous technologies. Plus in most cases the goal is not really destruction, but rather the threat of destruction. For example in the last movie, several EMP charges could have destroyed the majority of attackers. The machines know that, and they probably launched the attack with the goal of failing, but making humans more scared. The machines didn't foresee however that Agent Smith would be able to interfere and stop those EMPs.

So in summary, it's all a well-orchestrated plan to keep humans busy, and alive.

It's part of the allegorical elements of the film. By being powered by people, the machines are literally taking the power from the people (i.e. the opposite of 'power to the people'). So the machines are only existing because of the people's power, the people having sort of surrendered it to the machines. This is in addition to the in-Matrix subjugation humans also suffer.

The original storyline had the machines using the humans as a giant networked brain on the premise that the brain's processing power is far ahead of any hardware AI.

https://scifi.stackexchange.com/questions/19817/was-executive-meddling-the-cause-of-humans-as-batteries-in-the-matrix

Unfortunately the movie bosses considered this concept too highbrow for action / sci-fi audiences of the time. They thought we wouldn't get it, so the motivation of the machines was re-written (i.e. dumbed down) to have them using humans as a power source.

As others point out, this doesn't really stack up to a reality check, but really all they needed was a (vaguely) plausible reason for the matrix to exist. Using humans as batteries (or for the brain's supposed processing power) is a MacGuffin.

http://en.wikipedia.org/wiki/MacGuffin

After the success of The Matrix, the studio took even more creative control and essentially gutted the sequels of all philosophical content, turning them into highly stylised action films.

I think, within the movie's logic, it is because humans reproduce and therefore continually provide energy (ISTR that humans are even fed on human corpses) whereas nuclear reactors use finite energy sources.

My interpretation was that the body farms weren't used to generate energy, but to store it. i.e. the duracell battery that Morpheus holds up. If you have pulsed or unpredictable power sources, you need massive storage to match the supply to the demand. The leading fusion power concept today is the tokamak, which is a pulsed machine. Renewable energy sources are generally weather-dependent and hence unpredictable.

On the other hand I can't imagine that humans are very efficient for energy storage, compared to animals or any other inanimate technology. In that respect it seems more like a plot device.

My theory is that the Machines lack one thing that only Humans can have -> "Creativity".

The machines can maintain the status quo, but for any new technology development, they need human brains. Therefore, the machines keep all humans alive, to harvest their brain power for advancing the technology.

The machines creates the reality humans live. In exchange humans create the ultimate reality the machines depend to exist. The machines figured out that their artificial brain have no effect on the wave function and matter can never exist without the effect of wave function collapse caused by human brains. I hope this is the reason the movie creators had in mind. They would come up with a movie explaining how it was before the machines took over and as they exterminated humans the machines noticed that chunks of the material world also would disappear and until they fix this inconvenience they need the humans alive. Hope this is the reason as I would definitely watch the prequel!

Human-computation. The machines delegate problem solving / processing to humans, like micro-work/crowd-sourcing. See related SE question on this. Human-based computation is:

A computer science technique in which a machine performs its function by outsourcing certain steps to humans, usually as microwork. This approach uses differences in abilities and alternative costs between humans and computer agents to achieve symbiotic human-computer interaction.