The possibility of collateral RNA degradation poses a concern for transcriptome perturbations and therapeutic applications using CRISPR-Cas13. We show that collateral activity only occurs with high RfxCas13d expression. Using low-copy RfxCas13d in transcriptome-scale and combinatorial pooled screens, we achieve high on-target knockdown without extensive collateral activity. Furthermore, analysis of a high-fidelity Cas13 variant suggests that its reduced collateral activity may be due to overall diminished nuclease capability.