How do I check whether an OEKO-TEX or BSCI certificate is genuine?

Take the certificate number and the holder name and check them at the issuing body, not at the factory. OEKO-TEX certificates can be looked up in the OEKO-TEX public label database, and BSCI audit reports are verified through the amfori platform against the auditee ID.

Check the scope as well as the validity. A certificate may cover certain product classes or certain production sites and not others. A certificate that is genuine but out of scope for what you are buying is the failure mode that most often gets missed.