Best is to use an around-the-waist security belt for your passport, cash, and credit cards. The belts are basically a rectangular pouch with a velcro or zipper closure with an elastic strap that goes around your waist under your clothing. The belts are sold at luggage shops.
Carry in your purse or pockets (pockets are better) an ID that isn't your passport, and only cash & cards for the day. Keep everything else in the security belt.
If you don't want to use something like your driving license as your daily ID then get a passport card in addition to your passport. The cards are only valid for ground travel between the USA and Canada & Mexico, but are a good "government-issued photo ID":
http://travel.state.gov/passport/ppt_card/ppt_card_3926.html
I use a cigarette pack-size digital camera and carry it in a pocket. Trousers with cargo pockets work great, and I now rarely carry a day-pack.
If someone does steal my day-pack all they get is my lunch and maybe a map.
Send to yourself an e-mail message containing the information (account number, exp. date, contact number, etc.) from your passport, other ID, and any credit cards you plan to take on the trip. That way, the info is available from any computer connected to the internet in case something is lost or stolen. You can also scan the items and send the images to yourself.
That all said, I have traveled all over the world for decades and have had a bag stolen >> once <<. It was in Spain years ago and two men on a motorbike broke into my car for my carry-on bag. They saw me loading the car, followed me on the bike, and took the bag (after breaking a window) when I was only a short distance away. It happened very quickly. I no longer carry anything valuable in a bag.