The belts themselves contain vast energies - but I personally think near-orbital clutter should be minimised. Though a few units could sit in this belt and potentially deliver large quantities of energy. I don't think we should be putting all our eggs in any one basket, on a number of things not just energy production. We need lots of solutions - we will need lots of energy in lots of places(eventually) so it'd make good sense to setup a transmission network. As Earth requires energy too, excess can be easily offloaded. The largest problem with building something to operate in these belts I feel would be the sheilding required. How precisely was you planning to convert the radiation energy into thermal energy? Then what? Peltier effect? Stirling machines? combinations? something else?
Contamporary generation methods are possibly what is more likely to be deployed, at least short term, but it's definitely a good idea to be considering alternates. Solar panelling is "sufficient" for certain loads, if deployed in certain places. Obviously, the further you are from the nearest star the lesser the photon density - locally by the time you get out to about Jupiter the square foot per kilowatt becomes inconvenient. In a near Earth orbit location, like ISS, they use nearly 33,000 solar cells per "wing" which when fully extended is 35 metres (115 ft) in length and 12 metres (39 ft) wide. Combined (there are 8 wings) this generates between 84 » 120 kilowatts(obviously, in the shade of Earth this generates Zero and they rely on the batteries - currently Nickle-Hydrogen, being upgraded to Li-Ion(some installed)) These were unfortunately largely lifted before higher efficencies in the panelling was developed, so is likely between 30-40% efficencies on install date. In atmosphere it's commonly listed as 1% loss/yr - but is likely more - outside of atmostphere, the harshness of the sun effects more, so these are likely much more towards 80kW or lower by now(They don't need this much power, it was built overspec so the decay could still happen and yeild service life). Almost certain to require replacing by the time NASA abandon ISS in 2020»2025. Tl:Dr; Panels can be okay, in the right places, but then only for so long.