Depending on the type of fill valve what the remedy is.
Some older ball cocks have a washer while others have a diaphragm.
As the washer gets older it can start to disintegrate do to all the chemicals they are adding to the water supply (like chlorine) The diaphragm type like the Fluid Master depends on a rubber diaphragm type washer that can lose its elasticity.
Knowing the type of ball cock you have inside the tank you can then go to a plumbing supply and get the rebuilding kit for your type of valve.
I have found over the years it is much more cost effective to just replace the existing valve speaking from a contractors point of view. Considering a new 400A for example cost around $6.00 and for the peace of mind you have something new that should last many years (the labor rate varies considerably). You can try one more thing and that is to make sure the shut off valve by the tank is fully open then cracked (turned) back about a 1/4 turn so it doesn’t seize up in the open position. If all else fails hire a plumbing contractor.