If you are visiting a house during the week of mourning, you should be aware that this isn't a normal social call. The mourners aren't there to greet or entertain guests. Usually you don't ring the doorbell, or knock on the door, or even say hello to the mourner. The door is left open whenever there is anyone in the house.
Most Jewish mourners are too accustomed to providing hospitality to stop themselves from offering food to the people who have come to comfort them, but that is not their role. From the first ritual meal that the community sets before the mourners when they return from the burial, through the week of mourning, the community feeds the mourners. In synagogues, there is often a committee to coordinate cooking for a house of mourning, and sometimes friends arrange to send platters of food from a delicatessen instead of flowers. Food is traditional and flowers aren't. It's also traditional to give charity in memory of the person who has died.
If you aren't supposed to say hello and the mourner isn't supposed to jump up and offer you food, what do you when you visit a shiva house? The traditional words of comfort among Jews are "HaMakom y'nachem etkhem b'tokh sh'ar aveilei Tzion v'Yrushalayim"--"May God comfort you among all the mourners of Zion and Jerusalem." Or you can say "I'm sorry," or just reach out to the person physically with a handshake or a hug if they want that. It is enough to just be there and listen. Whether the atmosphere is dark and quiet with the mourners on low stools and the mirrors covered, or whether it is busy, with many people visiting and big platters of food, take your cue from the mourners.
The Guide to Death and Mourning for Interfaith Families is also available as a downloadable PDF and Word document.