To satisfy thirst:
* Satisfy
* Slake
* Assuage
* Allay
* Extinguish (figuratively, to put out a burning desire)
To extinguish a fire:
* Extinguish
* Douse
* Smother
* Suppress
* Put out
To cool or temper:
* Cool
* Moderate
* Dampen
* Subdue
Other:
* Still
* Pacify
* Calm
The best synonym will depend on the specific situation and the nuance you wish to convey.