Why don't modern spacecraft use nuclear power?

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.

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.

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.