Recent analyses of the cosmic microwave background (CMB) and the Lyman-alpha forest indicate a mild preference for a deviation from a power law primordial matter power spectrum (a so-called negative `running). We use an extension to the BAHAMAS suite of cosmological hydrodynamic simulations to explore the effects that a running scalar spectral index has on large-scale structure (LSS), using Planck CMB constraints to initialize the simulations. We focus on 5 key statistics: i) the non-linear matter power spectrum ii) the halo mass function; iii) the halo two-point auto correlation function; iv) total mass halo density profiles; and v) the halo concentration-mass relation. In terms of the matter power spectrum, we find that a running scalar spectral index affects all k-scales examined in this study, with a negative (positive) running leading to an amplification (suppression) of power. These effects should be easily detectable with upcoming surveys such as LSST and Euclid. In the mass range sampled, a positive running leads to an increase in the mass of galaxy groups and clusters, with the favoured negative running leading to a decrease in mass of lower-mass (M <~ 10^13 M_solar) halos, but an increase for the most massive (M >~ 10^13 M_solar) halos. Changes in the mass are generally confined to 5-10% which, while not insignificant, cannot by itself reconcile the claimed tension between the primary CMB and cluster number counts. We find that running does not significantly affect the shapes of density profiles of matched halos, changing only their amplitude. Finally, we demonstrate that the observed effects on LSS due to a running scalar spectral index are separable from those of baryonic effects to typically a few percent precision.
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