Or does capping a chimney always remove the ability to use a fire ?
The moisture is a result of lower flue temperatures allowing the water vapor in the exhaust gas to condense (dew point). A cap will reduce the flow, but does nothing to prevent condensation.
The only practical solution is to line the flue (with a metal liner) and remove the damper (if this was a wood burning fireplace, originally. The liner is sealed at the bottom with a fitting and an appropriate mortar. The liner is then attached to a cap for a end-to-end seal.
The water in question is acidic and will attack the mortar, potentially causing leaks and joint failure, over time