Most discussions of the Van Allen radiation belts focus on the necessity to shield sensitive electronics and biomass from radiation hazards. But are there any possible uses for this cache of highly energetic particles? For instance, can they be used as an energy source for spacecraft, or might it be possible to mine the antiprotons discovered in the belts?
There are a number of terrestrial uses for plasma, which could be adapted for space purposes; these could theoretically harvest the required plasma from the Van Allen belts. There are even a few terrestrial uses for plasma that have been suggested as future space technologies if some further development is made. Have a look into plasma windows (used in electron beam welding), they have the potential to be used as airlocks that hold the pressure differential but allow the passage of higher momentum objects - such as humans in space suits.
A second thought that springs to mind is fusion. Magnetic fields can be used to contain the plasma used in fusion, I would be hesitant to suggest that the plasma in the Van Allen belts has the right properties, but there may be a cross of with the technologies that can allow some form of energy exchange from the plasma to a spacecraft.
If your spacecraft were to pass through the south Atlantic anomaly (where the highest concentration of anti-protons has been found I believe), I would imagine you could mine the antimatter. Of greater concern would be storage and eventual return to Earth (if that was the intention of your question). Having antimatter held in a magnetic bottle brought back to Earth by reentry would not only be a scientific challenge, but the political implications of what could happen if something went wrong would be a massive barrier to any sort of mission like this.
A few links you might be interested in: