How do you get red wine stains from the bottom of a Ships Decanter?

You might want to consult a professional before doing this if it's an antique; a lot of people erroneously think that cleaning something will make it "worth more" when in fact it often has the exact opposite effect.

Assuming you want to proceed, you might try ethanol... a small amount of the highest proof grain neutral spirits you can buy. Let it sit, and gently swish it around (or, heck, get a bottle brush and scrub if you want, I don't care).