There are two main reasons why Brutus dominates Cassius before and after the assassination of Julius Caesar: (1) Brutus is established as the leader of the conspiracy, and (2) Cassius so loves Brutus that he blinds himself to the dangers of decisions made by Brutus.
1. Realizing that Brutus is well-respected and considered most noble in character, Cassius inveigles Brutus into leading the conspiracy to assassinate Caesar, thus providing Caesar's death with what the Romans will consider just cause since, as Brutus tells them, he "loves Rome more" and, as Antony later points out, Brutus was "Caesar's angel" and loved him. Cassius is also probably aware that he is known for being envious of Caesar and, if he were to lead the conspiracy, the Romans would condemn him. In Act I, for instance, Caesar himself recognizes the threat from Cassius, telling Marc Antony,
Let me have men about me that are fat,
Sleek-headed men, and such as sleep o' nights:
Yond Cassius has a lean and hungry look;
He thinks too much; such men are dangerous. (1.2.198-201)
Thus, with Brutus in the leadership role, Cassius knows that men will follow his orders as they hold him in a higher position of respect. In Act III, he says,
Ay, every man away.
Brutus shall lead, and we will grace his heels
With the most boldest and best hearts of Rome (3.1.132-134)
2. Cassius later displays another weakness besides his envy: Because he loves Brutus, he does not want to lose his friendship. In Act III, for instance, although he advises Brutus to kill Marc Antony and, certainly to not allow him to speak to the plebians--
You know not what you do. Do not consent
That Antony speak in his funeral.
Know you how much the people may be moved
By that which he will utter? (3.1.250-254)--
Cassius does not argue when Brutus fails to concur with his ideas; instead, he only says, "I know not what may fail; I like it not: (3.1.258). Later, in Act IV Cassius and Brutus argue; Cassius indicates his weakness regarding friendship in his words to Brutus as they argue and Brutus tells Cassius of his dislike for Cassius's faults. To this, Cassius replies,
A friendly eye could never see such faults. (4.3.96)
Then, they argue whether it is wise to move the troops to Philippi. Ironically, it is Cassius now who is superstitious and expresses his fears about the forthcoming battle. Logically, though, he contends that their troops should remain where they are and let those of the triumvirate exhaust themselves by marching. However, Brutus dominates Cassius with a reason that even sounds more like one of Cassius:
There is a tide in the affairs of men
Which taken at the flood leads on to fortune;
Omitted, all the voyage of their life
Is bound in shallows and in miseries.
On such a full sea are we now afloat,
And we must take the current when it serves,
Or lose our ventures.(4.3.244-250)
In his love for Brutus and sense of fatality, the more passive Cassius acquiesces,
Then, with your will, go on;
We'll along ourselves and meet them at Philippi (4.3.251)
Near the end of the play, Cassius asks his bondman Pindarus to watch and report what he sees because his own "sight was ever thick" (V.ii.21). Some scholars feel this physical myopia is symbolic of the clouded vision of Cassius regarding Brutus, whose friendship he values even above his own life. For it is this admiration and friendship which leads Cassius to acquiesce to Brutus even when his own judgment is better.