Katherine is Mathias's mom. Her main role is to warn her son to stay away from Barabas (should have listened to your mom, Matty) and, after his death, be distraught all over the place while vowing to avenge her son's death.
While not a bad lady, Katherine is, like the other Maltese Christians, narrow-minded and prejudiced. When she sees Mathias talking with Barabas at the slave market, she tells him, "Converse not with him: he is cast off from heaven" (2.3.158).
Katherine is one of the many minor characters of the play who, while they haven't really done anything wrong, per se, and are victimized by Barabas, still aren't completely innocent. Her cheerful bigotry may be justified in Barabas's case, but it's still a major problem for the hypocritical, deeply prejudiced society of Malta.