We find the answer to this question in chapter 10 of Robert Louis Stevenson's The Strange Case of Dr. Jekyll and Mr. Hyde.
This chapter narrates Henry Jekyll's final letter, where he discloses the truth about his dissociated personas. He also explains the reasons behind the actions of both, Jekyll and Hyde, and he somehow aims for some forgiveness, although he is aware of all the damage that his experiment has caused.
Fear is present in Jekyll's life not necessarily when he is first adopting the persona of Hyde, but when his experiment begins to get out of control. It is explained how Jekyll's moral and strict personality is actually weaker than Hyde's immoral and evil self. For this reason, Hyde's personality begins to take over that of Jekyll's and a transformation takes place where poor Dr. Jekyll is, literally, attacked from both ends and is left nearly impaired in the process. Hyde takes over Jekyll completely in the end.
This being said, we do not suspect fear in Hyde's mind except when it comes to saving himself from being discovered, or from being killed, such as it happened in the case of the child that he hurts.
An act of cruelty to a child aroused against me the anger of a passer-by, whom I recognized the other day in the person of your kinsman; the doctor and the child’s family joined him; there were moments when I feared for my life; and, at last, in order to pacify their too just resentment, Edward Hyde had to bring them to the door, and pay them in a check drawn in the name of Henry Jekyll.
However, fear as a driving force of actions only comes, again, towards the end when Jekyll (now slowly overpowered by the persona of Hyde) tries to make sense of what is going on. First, we find him hurting a woman who simply comes across his path when he is in an intense moment of thought.
He walked fast, hunted by his fears, chattering to himself, skulking through the less-frequented thoroughfares, counting the minutes that still divided him from midnight. Once a woman spoke to him, offering, I think, a box of lights. He smote her in the face, and she fled.
Fear is also the factor that leads Hyde to write those strange and insulting notes that are seen next to the textbooks that Dr. Jekyll would read. The hatred that Hyde felt as an evil creature begins to direct itself straight to Jekyll. Everything that Jekyll represents is resented and hated by Hyde. Yet, Hyde's fear of the situation, and of getting caught, is so profound that he begins to act irrationally.
The hatred of Hyde for Jekyll, was of a different order. His terror of the gallows drove him continually to commit temporary suicide, and return to his subordinate station of a part instead of a person; [...] Hence the apelike tricks that he would play me, scrawling in my own hand blasphemies on the pages of my books, burning the letters and destroying the portrait of my father...
Conclusively, Hyde only acts out of fear when he is about to get caught, be recognized, or is suspected of doing something. It is part of his hedious personality. When Hyde murders Sir Danvers Carrew in chapter 4, we know that Hyde is aware of Sir Danvers's link to Mr. Utterson. Even then, this particular murder is debauched and without reason. For Hyde's own sake, his actions are merely vicious. Fear only comes when he can no longer control a test gone wrong.