Judge Taylor obviously takes this trial very seriously. He appointed Atticus to defend Tom. It should have gone to Maxwell Green, but the judge made sure that Tom got the best defense lawyer to be had. He appointed Atticus for that reason. He knew Atticus would do all he could to give Tom a fair trial.
Although he does seem to be sleeping during session, he does always have his ear (not his eyes) on the trial before him. He does take the trial seriously, and his reaction to the verdict proves his resentment toward the jury's decision. Scout's narration mentions that Judge Taylor was "saying something. His gavel was in his fist, but he wasn't using it." Although we don't get to hear what he says there, he was obviously upset with their decision. The movie itself just has Taylor leave his seat and slam his door on the way out. Either interpretation shows his frustration with the verdict.
Judge Taylor appears to be rather old and tired during the trial. However, the reader knows that he must take his job seriously because he asks Atticus to defend Tom Robinson. As a county judge, he must be careful not to look too interested in the trial, or he might lost his job. He knows this but secretly goes to Atticus and convinces Atticus to defend an unpopular defendant. He knows Atticus will give Tom a fair defense. And he also knows that by appearing disinterested, he will also keep his job.
He takes this case very seriously and he knows that because of the prejudice found within the town, the people will make Tom guilty of a crime he has not done. The judge tells Atticus to take up Tom's case because he knows that this is the only way that he will be given a proper trial and maybe some people will come to understand the truth.
Judge taylor does takes this trial very seriously, as he has asked Atticus to defend Tom.