Non-standard analysis has been quite successful in settling existence questions in probability theory. Hyperfinite Loeb spaces allow for several constructions that cannot be done on standard probability spaces. In particular, NSA was quite useful for the construction of certain adapted processes. There is a paper by Hoover and Keisler, Adapted Probability Distributions, from 1984, in which the authors show that many of the properties that make hyperfinite Loeb spaces so useful where due to a property they called saturation: A probability space $(\Omega,\Sigma,\mu)$ is saturated if whenever $\nu$ is a Borel probability measure on $[0,1]^2$ and $f:\omega\to[0,1]$ a random variable with distribution equal to the marginal of $\nu$ on the first coordinate, then there exists a random variable $g:\Omega\to[0,1]$ such that the disttribution distribution of $(f,g)$ is $\nu$. An example of a saturated probability space that is not a hyperfinite Loeb space is the coin-flipping measure on $\{0,1\}^\kappa$ when $\kappa$ is uncountable. A relatively readable exposition of this approach can be found in the small book Model Theory of Stochastic Processes by Fajardo and Keisler. There are also several related papers and surveys on Keisler's homepage.
In a sense, we nowadays understand fairly well how certain powerful techniques of non-standard analysis work below the surface, so we can use a lot of the constructions freed of NSA. There isn't really anything where it is necessary to use NSA. Still, NSA is a rather poweful powerful and useful tool. A good overview over what it can do for probability theory, mainly the theory of sochastic processes, is in the article by Osswald and Sun in Nonstandard Analysis for the Working Mathematician by Loeb.
