I expect that as Keeper of the Grounds of Hogwarts, keeping the forest creatures at bay is one of Hagrid's tasks. Likewise, it's suggested in CoS that as Defense Against Dark Arts Professor, Lockhart is obligated to rid the castle of the basilisk. I would suspect the Care Of Magical Creatures Professor (typically not the same person as the groundskeeper) would play some role, too. In addition, one can't rule out the unspecified protective charms we're told keep Hogwarts safe.
Also, before Hagrid added Acromantulas and Grawp, there wasn't anything in the forest that wasn't manageable.
- JK appears to have changed her mind about whether Fluffy was released in the forest, probably realizing that that was, well, pretty idiotic.
- The centaurs keep to themselves, never go near Hogwarts, and seem to have a friendly relationship before Firenze defects. We're also told explicitly that they never harm children.
- The thestrals appear harmless and basically tame, Ministry designation be damned.
- Unicorns shy away from humans and are freaking innocence in animal form.
- Hippogriffs occasionally attack wizards, but only when they feel insulted.
- First-year Draco Malfoy says there's werewolves running around the forest, but that seems to be retconned out of existence in later books.
So assuming Fluffy and werewolves were awkward mistakes Rowling later rolled back, there's nothing in the forest that the highly-trained and capable wizards at Hogwarts couldn't handle. The fact that Dumbledore seems to have no qualms about sending small children into the forest for detention, and that Hagrid leaves them with nothing but a dog likely confirms that.
The Acromantulas were likely the only real threat, but I don't think Dumbledore ever even knew of their existence and Hagrid actually seemed to have them under control. But luckily, a year after Hagrid's control over them slipped, they were driven from the Forbidden Forest by Death Eaters and likely wiped out in the Battle of Hogwarts.