What advantage do quadratic basis functions have over gaussian basis functions in RBF implementations?
...a strange choice for a basis function, since distant points in a training data set would seemingly influence the results of a local solution more than nearby points. Why is this function used as a basis function?


The sum of their influences quickly fades away. It's well illustrated in 1d, where we can represent a spline either piece-wise, locally controlled, or as one, 'global' RBF sum with $|x|$ kernel (see here for example). This might seem strange as each $c_i|x-x_i|$ term is unbounded. Yet, together they cancel out automagically at a distance. You can see it by following the derivation at the link. Another example is a system of electric charges overall neutral.


  what advantages does it have over something like a Gaussian basis function?


Multiquadric and other globally supported RBF have a good hole-filling property: if you have holes that are irregular with unknown scales, they're going to be smoothly interpolated over according to surrounding trends from neighborhoods of corresponding scales. If your dataset has a regular sparsity of one radius you can estimate, you can use instead compactly supported RBF, or Gaussian RBF which is practically local.