Stuff balances by itself, if you provide worse service, less cargo or passengers will show up. If you provide good service more cargo or passengers show up.
For passengers, this doesn't quite work, there are just too many producers around in a city. Other options are building a passenger station somewhat outside town, so you have fewer houses around to provide passengers. Unless you have city expansion disabled, this will only temporarily help, as cities love to build houses in serviced areas
You can get passenger count down, but don't bother trying buses, you need long fast trains that are constantly loading on more than one platform.
The other options of course are to just ignore passenger counts at stations (they don't make sense anyway), or (what I do) totally ignore passenger and mail servicing.
For the default set, that's about it.
If you add NewGRFs there is a whole new world on options available. NewGRF industry sets have different mechanics for providing and accepting cargo, and passenger-fix GRFs exist as well (I never bothered to look at them, so no idea what exists there).
There should be a NewGRF list page on the Wiki to give you a starting point. When starting with NewGRFs, set-up only a few at a time for a game, to make it easier to understand what each GRF is doing.