Maybe keep door latched all the time and see what happens. BTW since they quit putting phosphates in the dish soap I throw a 1/2 of vinegar in when the rinse cycle starts.
Ants update: looks like they were getting in through the hinge of the door, I suspect there is a nest behind it (built in against an outer wall). Bought some traps from HD that have boric acid and sugar--supposedly they suck up the sugar and carry the boric acid back to the nest.