"Put on a creature you control" includes that creature entering the battlefield with -1/-1 counters on it. If a creature would enter the battlefield under your control with a number of -1/-1 counters on it while you control Vizier of Remedies, it enters with that many counters minus one.

If an effect puts Vizier of Remedies onto the battlefield with one or more -1/-1 counters on it, its effect won't apply. This is because you must control Vizier of Remedies before the creature begins to enter the battlefield for its effect to apply. The same is true of any creatures entering the battlefield with -1/-1 counters on them at the same time as Vizier of Remedies enters the battlefield.

If a creature you control is dealt damage by a source with wither or infect, that much damage is dealt, but one fewer -1/-1 counter is put on your creature. For example, if a 2/2 creature with lifelink and infect is blocked by Vizier of Remedies, the first creature's controller gains 2 life and puts one -1/-1 counter on Vizier of Remedies.

If multiple creatures with wither and/or infect deal damage to a creature at once, Vizier of Remedies causes only one counter fewer to be put on that creature. Its effect doesn't apply separately for each creature dealing damage.

Each additional Vizier of Remedies you control will decrease the number of -1/-1 counters put on a creature by one.