Enhancers are key drivers of gene regulation thought to act via 3D physical interactions with the promoters of their target genes. However, genome-wide depletions of architectural proteins such as cohesin result in only limited changes in gene expression, despite a loss of contact domains and loops. Consequently, the role of cohesin and 3D contacts in enhancer function remains debated. Here, we developed CRISPRi of regulatory elements upon degron operation (CRUDO), a novel approach to measure how changes in contact frequency impact enhancer effects on target genes by perturbing enhancers with CRISPRi and measuring gene expression in the presence or absence of cohesin. We systematically perturbed all 1,039 candidate enhancers near five cohesin-dependent genes and identified 34 enhancer-gene regulatory interactions. Of 26 regulatory interactions with sufficient statistical power to evaluate cohesin dependence, 18 show cohesin-dependent effects. A decrease in enhancer-promoter contact frequency upon removal of cohesin is frequently accompanied by a decrease in the regulatory effect of the enhancer on gene expression, consistent with a contact-based model for enhancer function. However, changes in contact frequency and regulatory effects on gene expression vary as a function of distance, with distal enhancers (e.g., >50Kb) experiencing much larger changes than proximal ones (e.g., <50Kb). Because most enhancers are located close to their target genes, these observations can explain how only a small subset of genes - those with strong distal enhancers - are sensitive to cohesin. Together, our results illuminate how 3D contacts, influenced by both cohesin and genomic distance, tune enhancer effects on gene expression.