We study the evolutionary dynamics of an asexual population of nonmutators and mutators on a class of epistatic fitness landscapes. We consider the situation in which all mutations are deleterious and mutators are produced from nonmutators continually at a constant rate. We find that in an infinitely large population, a minimum nonmutator-to-mutator conversion rate is required to fix the mutators but an arbitrarily small conversion rate results in the fixation of mutators in a finite population. We calculate analytical expressions for the mutator fraction at mutation-selection balance and fixation time for mutators in a finite population when the difference between the mutation rate for mutator and nonmutator is smaller (regime I) and larger (regime II) than the selection coefficient. Our main result is that in regime I, the mutator fraction and the fixation time are independent of epistasis but in regime II, mutators are rarer and take longer to fix when the decrease in fitness with the number of deleterious mutations occurs at an accelerating rate (synergistic epistasis) than at a diminishing rate (antagonistic epistasis). Our analytical results are compared with numerics and their implications are discussed.