Giving your monster a healthy dose of Fear involves taking a strong hand with it whenever it tries to slack off, keeping it on a rigorous Training schedule, regularly feeding it food that it dislikes and being inattentive to its emotional and physical needs (Stress and Fatigue).
In some games, monsters will develop a nasty Trait called ______ that will cause it to have a breakdown during battle. In other games it ensures that you cannot raise your monster's Loyalty stat beyond 50 points, because it believes you to be too harsh. This concept is explored in the anime series, where Allan instills fear into his Worm, while Genki spoils Mocchi.
You can balance out a Fearful monster by Spoiling it every once in a while, giving it gifts and being kind and fair in your choices. This will give you a balanced Raising Style.
Not to be confused with a Battle Effect in Monster Rancher 4 also called Fear.