Why don't modern spacecraft use nuclear power? The Voyager 1 & 2 spacecraft launched in 1977 with Plutonium as their source of electricity. 34 years later they claim these two spacecraft have enough power to last them until at least 2020. That means they'll have had enough power to last them at least 42 years. It obviously offers enough power to literally send transmissions across the entire solar system.
Why don't modern spacecraft use nuclear power if it offers such longevity and power? You would think that 34 years later we would have the technology to make this even more viable source of electricity then when the Voyagers were designed and built. The New Frontiers spacecraft seems like an excellent candidate for nuclear power.
 A: The fact as I see them:


*

*RTG's are not dangerous to launch.  In a CATO, you would likely get your RTG back, probably in working order.

*The radioactive material is not that dangerous even if lost (and lost means off Cape Canaveral, not Los Angles...I don't believe there are any launch facilities that have a worry of hitting a population.

*In terms of power production for a major spacecraft, we CANNOT do it with a single launch and solar power.  Cassini would have required 1400 KG of solar panels (losing power in umbras and requiring even more mass in undependable batteries,plus requiring more heat energy) vs 168kg for it's RTG.  No comparison.

*For destination-bound manned spacecraft, shielded rtg's will actually PREVENT radiation exposure due to the decreased travel time afforded by more delta V.  They are more dependable due to extreme low complexity compared to a deployable solar array (not to mention continuous power availability).

*PU-238 is a strong Alpha emitter, and emits little else.  Alpha particles are easy to shield, as they do not even penetrate your skin.  Typically, pu238 RTG's do not require shielding other than their own functional case, and you have to eat the stuff to get sick.
My conclusions:


*

*Solar is not used due to expense or expediency, it is used due to political pressure afforded by general uneducated paranoia.

*The same opinions have resulted in loss of breeder reactors (and therefore loss of PU-238 production).  Our actions in this arena have been like trying to protect teenagers from pornography, and just as self-defeating.

*Lack of RTG use makes serious space exploration extremely difficult.

*Agreeing that RTG spacecraft should not be used in orbit or be allowed to re-enter is an acceptable compromise.
Reference material:  http://fti.neep.wisc.edu/neep602/FALL97/LEC22/lecture22.html
A: It's all a question of if they need it.  Most that are staying within a couple AU of the sun can get sufficient power from solar panels.  It's when they start getting further away that they use an RTG.
For example, New Horizons, which launched in 2006 (which is considered to be 'modern' when you only launch a few probes per year) is going to Pluto, so it won't be able to get sufficient power from solar panels, and uses an RTG.
Like anything else, it's a question of risk and cost.  If it's cheaper, or lower risk without significantly increased cost, they'll go with the alternative.  
A: The case of neither American nor Russian deep-space missions
No one mentioned in the answers the “specific” case of deep space missions which are neither American nor Russian (which as of 2017 means Japanese, European or Indian missions. China will likely be soon (2020?) added to the list). These space agencies essentially have not developed the RTG technology, even if they are all linked with nations developing nuclear reactors and (except Japan and most, but not all, ESA members) weapons. For the ESA at least, this lack of development is clearly political, since its constituting member states have strongly diverging views on nuclear technologies, and the most “nuclear states“ are likely reluctant to share their know-how.
A specifically interesting mission to look in this solar vs nuclear power debate in deep space is the European Rosetta/Philae mission, which was solar powered. Fitting the mission with a RTG would have been politically impossible, even if the ESA had access to this technology. They developed more efficient solar panels instead, as explained on the mission’s FAQ.
This choice had a strong effect on the mission of the Philae lander:
following a “chaotic” landing, it essentially ended in the shadow of a cliff on the comet 67P/Churyumov–Gerasimenko and couldn’t easily recharge its battery,
not matter how efficient its solar panels were.
This severely limited his science mission. Had it be RTG-powered, the effect of its “strange” landing would have been much less important.
A: One big issue with nuclear power in space is that you need to discard the heat somehow, which for RTGs you can only do by radiating the heat. 
You end up having heat-radiating panels in place of solar panels, with substantially lower energy output per kilogram than solar panels, unless you are very far from Sun. Only the spacecraft that go very far from sun, or spacecraft that have to operate in the shadow for extended periods of time, or some landers, use RTGs; that is true of modern spacecraft as of the old spacecraft. The Cassini probe uses RTGs, the Curiosity rover (launched in 2011) uses RTGs. The premise of question is simply untrue.
The nuclear lobby loves to talk how everyone fears nuclear power for no reason, and makes up things like the one prompting your question, I guess that's where the disinformation originated. The fact is, nobody stopped using nuclear power in space. Some minor niches (the nuclear power for operating in the shadow when orbiting Earth) may have been lost to improved solar panels, batteries, and kinetic energy storage; the failure probabilities for the space launchers are more accurately known, which may also have resulted in adjustment for the cost-benefit calculations involving the spacecraft loss.
A: The real problem with RTGs is that the US stopped making Pu238 in the 80s and has been very slow to start up production again, purchasing all our spacecraft Pu238 from the Russians (who have now also run out). I don't know about the byproducts from the breeder reactors, but Pu238 itself is actually not that dangerous to handle, and only toxic if ingested.
A: edit: I originally had some points about the inefficiency of RTGs, but after some more research prompted by @Jeremy I found that it's not really a valid point when they're used appropriately for the spacecraft's mission. The RTGs used by Galileo at Jupiter generated 300W of power, whereas the solar panels that will be used by Juno at Jupiter will generate 450W of power. Solar arrays are also much larger and heavier than RTGs and impact the delta-V budget of the spacecraft, a costly interaction. The reason that solar arrays are used in some spacecraft are outlined in the points I make below so the efficiency factor doesn't really come into play.
Radioisotope Thermoelectric Generators (RTGs) are used when a spacecraft will be venturing too far from the sun to get enough power from it, or when it experiences extended periods of darkness while still needing to operate. This is the case with the Pioneer missions, Voyager missions, the Cassini missions, as well as the science experiments left on the moon during Apollo, and surely more that I haven't thought of.
RTGs are dangerous, especially if the spacecraft fails during launch, or an earth-flyby goes badly (this could spread radioactive material across a continent), and don't generate much power when compared to solar panels in close proximity to the sun.
Solar panels are used for missions that will almost always have a clear view of the sun, where they can generate much more power than RTGs can.
A: One aspect is the concern for if the spacecraft were to fail to launch correctly and ended up crashing back to earth. In such cases, the nuclear radiation pollution could be severe if it ended up crashing in inhabited areas.
A: Since no one's mentioned it yet, I will: Apollo 13 carried a standard science payload which included an RTG originally intended for deployment on the moon and (I presume) not particularly well shielded. This meant that some additional care had to be taken when deciding where to dispose of the lunar module, as well as additional PR stress on NASA which I'm sure left its imprint. As I recall Andrew Chaikin treated this quite well.
(If someone else has a better understanding of the story please chime in!)
A: Some do: "Cassini is powered by 32.7 kg of Plutonium-238".
A: We do, but not everywhere. The typical plant is the thermonuclear type, using plutonium. to create the kind of power for the ISS would be about $500M. the solar array is like $10M. there is the danger as well. the probes that use Pu are unmanned and of zero risk to people. if the iss lost orbit there would be a big mess.   
