When you head out on a mission, you're putting your fortune at risk. You can place money in a safe if there's a risk that you'll meet an untimely end. Missions increase that risk, so you should make a habit of depositing most of your walking around money in the safe before you try to move the story forward that next step. Then, if you fail the mission and lose half of the money currently on your person, the loss is less of a disaster.