Firstly, there is the traditional concern of the green movement with nuclear waste and nuclear weapons proliferation. Nuclear advocates say these concerns are misplaced because of nuclear technology innovations (although its not clear how close the new technology is to wide deployment).
Secondly, there is an economic and free-market concern. Nuclear power plants generally only get built in highly regulated, centrally planned electricity markets with a great deal of Government financial support.
The projects to build them, at least in modern Western states, tend to be complex, expensive and prone to overruns. These projects generally need the plants to run all the time with guaranteed rates for the business case to stack up.
This is entirely at odds with solar and wind, which are highly democratised -- your Aunty can put solar on her roof -- and even when deployed at 'grid scale' tend to suck the profits out of wholesale electricity markets, because they have the lowest short-run costs and always out bid other energy sources.
In deregulated electricity markets renewables plus gas beat nuclear on price. We need storage to push gas out of the market.