How should I state my peer review experience in the CV?

This is the sort of thing which will vary more from field to field, and you may want to talk to other people in your field, but my general inclination is not to do so. Unless you have been specifically told that there is some review board you are on, or been invited to be on, this may not even exist. Aside from the ethics, if this is a prominent journal, and one of the editors sees your CV, they could react extremely negatively. A better solution might be that if you are going to list what journals you have reviewed for is to list how many articles you have reviewed for a journal if you have reviewed more than one.

Can I use for instance, "member of the review board" for journals X and Y?

I agree with Joshua, it's a bit risky to present it this way unless you are sure that this journal has something that they call "review board" and that you are on it (it might not exist or have a different name). Personally I just have a section called "Reviewing activities" with a list of journals/conferences I reviewed for ordered by year.