The CA payment vouchers tells you to include the SSN on the check, so if clients ask me if they should include it I generally tell them yes.

Ive had so many people say "But I don't want them to see the SSN on the check", ummm, only the tax agency and the bank are going to see it and they both already have your SSN.....if it gets lost in the mail, the payment voucher with your SSN is right there with it, so its hardly a secret.