Judge Taylor is a laid-back judge from Scout's description of him. He often has a cigar that he chews on while he listens to the trial, and people sometimes suspect him of falling asleep. However, when a lawyer tested the sleeping theory, Judge Taylor proved that he was listening and pay attention the whole time.
Judge Taylor is certainly serious about this being a fair trial and about there being order in his court. He continues to tell the spectators they need to be quiet or they will find themselves in contempt of court. He makes Bob Ewell "behave" on the stand and answer the questions, and he comforts Mayella so she is able to give what he hopes is truthful testimony. The most important piece of information that proves Judge Taylor takes the trial seriously is that he appoints Atticus, the best lawyer in area, to be Tom's lawyer.
I think that the judge takes it seriously but not at first because he is just like the people on the jury