Finding balls that contain a point
Note that point location in an arrangement of balls (of different radii) is equivalent to point location in an arrangement of hyperplanes one dimension higher by using the usual paraboloid lifting map $(x_1, \ldots, x_d) \mapsto (x_1, \ldots, x_d, \sum_i x_i^2)$. If you call this lifting map $\Phi$, then it's easy to see that $p \in B$ if and only if $\Phi(p)$ is above $\Phi(B)$. 

Now you have the problem of point location in an arrangement of hyperplanes (note that once you have such a structure it's usually easy to annotate each cell with the list of hyperplanes it is above). That problem can be solved by a variety of methods: for example there's an algorithm by Meiser that runs in time $O(d^5 \log n)$ but requires $n^d$ space. It's not clear from the question which is more important: space or time...