Box allows you to password protect shared links set at "People with the link", requiring the people accessing the link to enter the set password to access the file or folder. This is a premium feature which is unavailable on free personal accounts.
To use this feature, click on the little blue box with the white downward arrow next to the file or folder, hover over "Share". This will open up a pop up window.
On the pop-up window, under the "Advanced" settings, click the "Require Password" section. Enter a password in the text field and click the checkmark.