Does Steve's race play a role in his trial in Monster?

It might. While it never explicitly comes up during the trial, O'Brien, Steve's lawyer, thinks that it does. She says:

Well, frankly, nothing is happening that speaks to your being innocent. Half of those jurors, no matter what they said when we questioned them when we picked the jury, believed you were guilty the moment they laid eyes on you. You're young, you're black, and you're on trial. What else do they need to know?

She makes it clear that race is a factor in how the jury sees Steve, and it is that factor—unconscious bias—that makes it so the jury only sees Steve as guilty. Race is a powerful construct in our society, and the idea that a trial could happen where it doesn't come into play would be a fantasy. The novel is fiction, but Myers attempts to set it up realistically, meaning that the jury considers Steve's race even if they don't talk about his race during the cross-examinations or with witnesses.
Ultimately, it makes sense that Steve's race would play a role in the trial. His race is a handicap in the trial, like O'Brien says, because it denies him the right to being considered innocent until proven guilty. During the entire trial, Steve's lawyer works to help the jury see him in a positive light, mostly because he is not fighting to prove he didn't participate in the murder but rather to disprove the stereotypes and racism that the jury had already used to judge him before the trial even started.