Is it permissible to financially assist one's husband in obtaining an English language certificate through unlawful means, and would this be considered participation in sin?

We understand from your question that your husband wants to pay money to the institute director to obtain a certificate without an exam or passing, and this is not permissible. It is cheating and deception, and the money paid is a bribe to obtain something that is not rightfully due. Therefore, it is not permissible to assist him in this, and you should advise him to follow the legitimate path. However, if he is entitled to the certificate and finds no means other than paying money, then it is permissible for him, and it would be a bribe for the recipient. You are not obligated to assist your husband with your money unless it is willingly.