Cloud storage and edge computing are utilized to address the storage and computational challenges arising from the exponential data growth in IoT. However, data privacy is potentially risky when data is outsourced to cloud servers or edge services. While data encryption ensures data confidentiality, it can impede data sharing and retrieval. Attribute-based searchable encryption (ABSE) is proposed as an effective technique for enhancing data security and privacy. Nevertheless, ABSE has its limitations, such as single attribute authorization failure, privacy leakage during the search process, and high decryption overhead. This paper presents a novel approach called the blockchain-assisted efficient multi-authority attribute-based searchable encryption scheme (BEM-ABSE) for cloudedge collaboration scenarios to address these issues. BEM-ABSE leverages a consortium blockchain to replace the central authentication center for global public parameter management. It incorporates smart contracts to facilitate reliable and fair ciphertext keyword search and decryption result verification. To minimize the computing burden on resource-constrained devices, BEM-ABSE adopts an online/offline hybrid mechanism during the encryption process and a verifiable edge-assisted decryption mechanism. This ensures both low computation cost and reliable ciphertext. Security analysis conducted under the random oracle model demonstrates that BEM-ABSE is resistant to indistinguishable chosen keyword attacks (IND-CKA) and indistinguishable chosen plaintext attacks (IND-CPA). Theoretical analysis and simulation results confirm that BEM-ABSE significantly improves computational efficiency compared to existing solutions.