How do I know if my vermouth has gone bad?

Smell and taste it. Fresh vermouth smells bright, slightly floral, slightly bitter; oxidized vermouth smells sharp, vinegary, and one-dimensional. If you opened the bottle more than 60 days ago and didn't refrigerate it, it's almost certainly oxidized.

Vermouth is fortified wine and oxidizes after opening, even refrigerated. Tell-tale signs of oxidation: (1) Sharp, vinegary note on the nose. Fresh vermouth smells layered — herbal, slightly sweet, slightly bitter. Oxidized vermouth has a one-note tartness that overpowers the herbal complexity. (2) Darker color. Sweet vermouth deepens from ruby to muddy brown over time. Dry vermouth darkens from pale straw to deep amber. (3) Flat taste in cocktails. A Martini suddenly tastes harsh; a Manhattan suddenly tastes thin. Both are vermouth telling you it's done. Timeline rules: opened dry vermouth, refrigerated — 30 days. Opened sweet vermouth, refrigerated — 45 to 60 days. Opened anything, room temperature — 7 to 14 days before noticeable decline. Don't worry about 'spoiled' as in unsafe; vermouth at 16 to 18 percent ABV is too alcoholic for harmful microbes. Just stale, and stale vermouth makes bad cocktails. Buy smaller bottles (375 mL) if you use vermouth slowly; you'll waste less and have fresher pours.