Project Nemesis is a fan driven website for games that use the One-Roll Engine (like Nemesis, Wild Talents, Reign and Monsters) or Chaosium's Basic Roleplay System (BRP) (like Call of Cthulhu) and the Delta Green setting.
The usefulness of a safe house / green box is dependent on its location. A unoccupied safe house in a bad part of town would definately get broken into and occupied by local crack heads. The same thing could happen in the burbs, of course, although I think partying teenagers would be more likely. How partying teenagers differ from crackheads is left as an exercise for the reader.
A storage facility kitted out as a safe house seems like it would have a lesser chance of being broken into, depending on locality.
A hotel room, permanently rented, might arouse some suspicion from the staff. I'd expect locally owned hotels ( as opposed to national chains ) would be preferred. A smaller staff that have a more significant investment in keeping the hotel a going concern would be preferred, since there's a smaller pool of people to leverage with threats of IRS audits, sizeure of the property via drug laws, etc. . Of course, this would mean revealing that the room was rented to staff, which might not be good... although could work for DG as aflase flag.
Offices kept as safe houses would be good. Make it an IT consulting company and people comming and going at all hours seems normal. Need medical supplies? Dentists office.
As to who shows up on the documents as owner, DoJ would likely invent an identity (either corporate or individual ), but among the standard problems with that ( no record of existance before a certain, recent, date ) there is a chance the idenity is a real one: http://www.nbc4i.com/news/4507402/detail.html
DG might use the names of dead agents; since the cross referencing of birth and death records is difficult at best this would likely go undetected.
Better still, a locally owned company purchases the property. The locally owned company is in turn owned by a delaware or nevada incorporated company, which is in turn a wholly owned subsidiary of a company incorporated in the bahamas, which is in turn owned by a company from the isle of man, which is in turn owned by a company chartered in the netherland antillies, which is in turn owned by Alphonse.
Friendlies can also play a large role here; no better cover than a legitimate owner who lets the agents use their property. In the case of the hotel owner above, best for safe houses might be a husband / wife couple who own and/or run a hotel.
Less good would be a vacation home, since those are in remote areas (although seaside locations seem perfect for DOs ).
Worst would be a suburban home with a spare bedroom and screaming teenagers around. Agents come back to the safe house to find the parents absent and little timmy trying to shoot the cat with their MP5.