Train yourselves first to shut doors - anyone who has kids will understand that one!! (My husband has a thing about switching off lights when you leave a room - I am all for it, save money and energy - but he is totally obsessed! I am still the one who has to follow him around turning off his lights as he follows the kids turning off theirs!!!)
It is a lot more serious than it sounds. When I worked for Guide Dogs we had to take a dog out of service because of this habit. The owner was a male who insisted that he lived on his own at all times and couldn't understand what was happening but the third time the dog was rushed to the vet to have a pair of tights (pantyhose?) removed from it's gut was one time too many!!
Lid the laundry basket, tidy up the clothes on the floor (thank you Paddy for never showing this inclination - I could never manage that one!!!!) and consider putting a few mouse traps on top of the laundry. They won't be able to hurt the dog but may just make him think twice about going there again!!
Try using a bungy cord across the top of the basket - you know, one of those elastic rope type things that has a hook at each end - used to strap things to the roof of cars - it should stop the dog getting the lid off.