I hate people who try to get where I've gotten with falsified credentials.
One of the problems is that it sounds like he/she is already there, on the BOD, an officer.
Questions do need to be asked, but a lot depends on the situation.
How long have they been there and what is their record? If they've been there a long time (2 years or more), with a solid record, then I would consider it a forgivable offense. If they're there without some prior track record, or with a poor record, then it's clearly time to remove them.
Fundamentally, you have to take a hard look at this person's character. If they are a person with a solid character, who made a mistake, and they do good work, then I'd say forgive.
Otherwise, you need to start looking at what else may be falsified in your organization. It may turn out that money they managed is missing, or that decisions they have made were unethical or worse, illegal. The investigation needs to go beyond just the credentials to everything they have done since getting there. Not fun, but necessary, to find the possible "damage".
Experience tells me they probably don't have that great of a character if they're willing to use false credentials to get the job.