I understand there are monsters quite far down if you keep digging. Can they get past doors?
Doors can stop regular invaders, when the doors have been set to "forbid passage". Those doors can still be opened by thieves, who can pick the locks. Be careful though, because there are some monsters designated as "building destroyers" who have the ability to destroy doors entirely.
The best way I've found to stop things from getting in is with constructed walls. As in, breach the fun areas, then very quickly wall off that area until you are ready to deal with whatever you've found. Be quick with the walling though.
For more about doors, look here: https://dwarffortresswiki.org/index.php/DF2014:Door
For more about building destroyers: https://dwarffortresswiki.org/index.php/DF2014:Building_destroyer