Other answers have provided general ideas about how to confirm that individual sources are distant galaxies and not clusters, so I'll focus on the question of how astronomers in the Dark Energy Survey (DES) might be confident about galaxy identifications for the hundreds of millions of sources in their catalogs -- particularly as the overwhelming majority of those sources do not have direct distance measurements or spectroscopy.
I'm going to concentrate on globular clusters (GCs), since these are the ones that might be confused with galaxies (they are bright and can be found outside the interiors of nearby galaxies). The simple answer is that GCs are small, and thus won't be mistaken for galaxies unless they are quite nearby. But the number of nearby GCs is vastly smaller than the number of objects classified as "galaxies" in the DES catalogs, so there will be very, very little confusion.
The identification of "galaxies" in the DES -- as described in the Data Release 2 paper -- is based on whether or not they are big enough (in angular size) to not be stars. Stars (and also active galactic nuclei) are point sources, and will be "unresolved", having the same size in the images as the point-spread function (PSF) of the observations (full-width-half max $\sim 1$ arc second for DES). Galaxies, being extended objects, will be somewhat broader in the images (they will be "partially resolved"). As the paper describes, they have a classification for each object; there are $\sim 543$ million objects with "EXTENDED_COADD" $> 1$, which they refer to as a "benchmark galaxy" classification.
GCs are small -- a typical half-light radius (the radius within which half the total stellar light is found) is about 4 parsecs. (Bright galaxies, on the other hand, have half-light radii of 1000 parsecs or more.) This means that they will become indistinguishable from stars once they are more distant than, say, 5 or 10 megaparsecs.
So only GCs that are close enough to be partially resolved (but not so close they are actually resolved into individual stars) could be mistaken for galaxies, and this means there will only be a few thousand or tens of thousands of possible cases. Given that DES has hundreds of millions of partially resolved sources in their catalogs, you can be pretty confident that most of those are not GCs.
Moving forward, it's possible to discriminate between galaxies and the few partially resolved GCs via photometric redshifts. This is an approach that uses the brightness in different filters (DES uses five) to estimate the redshift of a source. (This is inferior to using actual spectroscopy, but doesn't require time-consuming additional observations.) GCs can then be identified as sources with a redshift of $\sim 0$, unlike the vast majority of galaxies in the catalog.