The determinateness problem, i.e., deciding whether a given multi-sequence $(m_k)_{k \in \mathbb Z}$ indeed comes from any measure, is a well known problem for 1-dimensional case, but the multidimensional case still attracts fresh research. A nice recent paper on this problem is by Blekherman and Lasserre . It seems that in your case, however, that is not the issue: you know that you have (truncated) moment data of a measure (that presumably has a density?), and want to reconstruct its density.

From the practical perspective, there has been a considerable "excitement" about entropy optimization methods, which was advocated heavily by Jaynes , in signal processing and applied math communities at the beginning of 80s. In short, these methods treat truncated moment data as constraints on optimization of an entropy functional, yielding ansatz for the reconstructed density to be, for 1D, $\rho(x) = \exp \sum_{n=0}^{N-1} \lambda_n x^n$ for $N$-truncation of the moments.

Good overview is given by Borwein. You might also take a look at papers of T. Georgiou, who works in applied settings and Lasserre.

The maximum entropy approaches have given considerable success in physics literature, but they do have their numerical downsides, which mainly stem from numerical sensitivity when higher number of moments is used. I believe the first approaches by Mead and Papanicolaou handled perhaps 9-13 moments, before the round-off errors kills the convergence of the method. The way around is to take trigonometric moments, instead of monomials, which basically constrains the density by its Fourier coefficients instead of power moments. An excellent survey of numerical issues was given by Abramov who also provides a maxent toolbox on his website (all the way down).

The Stieltjes-Cauchy transform you mentioned brings the moment problem into the complex domain, by producing the function $F(z) = \int (z - x)^{-1}d\mu(x)$. Its power expansion is the generating series for moments of the original measure $d\mu = \rho(x) dx$ on the real line. Conversely, if one is given a Stieltjes transform $F(z)$, taking a certain limit evaluates the density of the measure pointwise $\lim_{\epsilon \downarrow 0} F(x + i\epsilon) - F(x - i\epsilon) = \rho(x)$. Generalizing Stieltjes transform to higher dimensions is non-trivial, as it can be done in more than one way, to yield, e.g., multivariate Stieltjes-Cauchy transform (where the kernel is just a product of Stieltjes-Cauchy kernels along each dimension), or Fantappi`e transform, where kernel contains an inner product of coordinates $(z_0 + z \cdot x)^{-1}$.

However, I have no knowledge about infinite-dimensional domain. All that I've written applies to finite-dimensional (in certain cases, compact) supports of measures. I would be interesting in seeing other answers on infinite-dimensional setup.