This depends on your contract with said client. It also somewhat depends on the situation and size of error. But in general: NO, if you provided a proof and gave a review and client says send to print then the client has signed the document off. They have approved, and that person takes responsibility.
There is a extremely high chance of error in any moderately sized document. Even professional copy editors can not guarantee error free results. Let alone spot errors incurred when the document was transcribed, it incurs doubling of chances for error.
The reviewer is responsible
The purpose of giving out a document for review is for the reviewer to check for errors. Therefore the reviewer is responsible for their failure to review. If theres no binding consequences for reviewing then there is a chance that the reviewer lest a n error slide only to ask for a rebate later. Even in the case where its clear that its not in clients interests to cheat its still the responsibility of the reviewer.
Your contract should state this as a fact. It is sad but most people do not take this kind of responsibility seriously. Once its sent for manufacturing its done with, no editing. Bills have to be paid for etc. Be sure to remind your clients of this.
You are responsible to a part but you should be shielded because it would make it impossible for you to operate under the assumption of perfect. However there are situations that might make you liable.
You are at fault when:
- Client notified you of the problem you failed to address the problem.
- For some reason versions got mixed up so that the error popped back to existence.
In other cases you should exercise your judgement.
Hire a external editor to handle the checking. Personally i would never ever transcribe the text myself, i would send it out to somebody else or have the client transcribe it themselves.
Update your base contract template now if you do not have a stipulation in your contract already.
Also take the advice of @go-me: Never type stuff. People get so upset about typos and they are nearly impossible to wean out. Only very few persons can actually spot all errors, beforehand. And it is very expensive to have error free stuff.