Friends of the Smithsonian can find their Member Number on their Member Card here:

James Smithson Society Members can find their Member Number on the back of their Member Card here:
Friends of the Smithsonian Member Card
 James Smithson Society Member Card

Castle Circle Members can find their Member Number on the back of their Member Card here:

Castle Circle Membership Card

Or, if you are a Friends of the Smithsonian member and have received a renewal notice in the mail, you can find your Member Number here:

FS Renewal Form