The outcome of species interactions in a variable environment is expected to depend on how similarly different species react to variation in environmental conditions. We study community stability (evenness and species diversity) in competitive communities that are either closed or subjected to random migration, under different regimes of environmental forcing. Community members respond to environmental variation: (i) independently (IR), (ii) in a positively correlated way (CR), or (iii) hierarchically, according to niche differences (HR). Increasing the amplitude of environmental variation and environmental reddening both reduce species evenness in closed communities through a reduction in species richness and increased skew in species abundances, under all three environmental response scenarios, although autocorrelation only has a minor effect with HR. Open communities show important qualitative differences, according to changes in the correlation structure of species' environmental responses. There is an intermediate minimum in evenness for HR communities with increasing environmental amplitude, explained by the interaction of changes in species richness and changes in the variance of within-species environmental responses across the community. Changes in autocorrelation also lead to qualitative differences between IR, CR and HR communities. Our results highlight the importance of considering mechanistically derived, hierarchical environmental correlations between species when addressing the influence of environmental variation on ecological communities, not only uniform environmental correlation across all species within a community.