Need to caulk between mount and exterior wall?
If the fence is plastic there should be no need to worry about water damaging any of it's pieces.
If the rail is removed and there's a chance of water getting through to the house it is a good idea to caulk the top and sides of the rail. Urethane caulk is durable and paintable, but very hard to tool (very sticky). 100% silicone is the most water proof, but not paintable. Latex will work get a 40 year-type for exterior use.
If the fence is attached to a section of the house that is constructed of wood or may allow water to enter the house, than yes caulk away.