Buy a cover. Passport covers will help prevent damage that can happen while you are carrying it. Continuous use can wear the edges of a passport, and water damage from rain or sweat can destroy the pages. A cover is a good investment to protect your passport from wear and tear.
Wear your passport around your neck or waist. Special travel carrying cases will allow you to wear your passport in a zipped passport wallet that is worn around your waist or your neck. Both cases are slim enough to be worn underneath your clothing, so it can't be easily seen or reached by pickpockets. A carrying case will also be useful when traveling through customs and boarding your plane.
Make a copy and store it safely in your room. If you are not traveling across borders or checking into hotel rooms, it might be a good idea to make a copy of your passport to carry with you and lock the original in your suitcase or hotel safe. Many countries consider copies acceptable while just walking around. If you have a visa, make a copy of the visa as well.