Diagnosis
The O-ring or washer inside the spigot is worn out or has mineral grit on it. The Berkey spigot is a simple mechanical valve, and that seal degrades.
How to Fix It
Unscrew the spigot from the outside of the lower chamber (have a towel ready). Disassemble the spigot by unscrewing the nut behind the handle. Inside, you'll find a small rubber O-ring or flat washer. Clean it and the chamber it sits in. If it's flattened or cracked, you need a replacement. Berkey sells spigot repair kits with new O-rings.
If That Didn't Work
The brass stem inside the spigot itself could be pitted or corroded, preventing a good seal even with a new O-ring. In that case, you need a whole new spigot assembly.
Heads Up
When you reinstall the spigot, don't overtighten the locking nut against the chamber wall. Hand-tight plus a quarter-turn with pliers is plenty. Too much force can crack the stainless steel nut or the chamber's plastic threading.