Writing a set of JavaScript conditionals more efficiently
I like @janos solution, but if that is not possible for you, you might extract the code into a function. I know, it's just one line of code, but I think it would still be beneficial:

// marks className element as complete if isEdit is true
function addComplete(isEdit, className) {
    if (isEdit) {
        $(className).addClass("complete");
    }
}

addComplete(stepOneEdited, stepOne);
addComplete(stepTwoEdited, stepTwo);
addComplete(stepThreeEdited, stepThree);
addComplete(stepFourEdited, stepFour);
addComplete(stepFiveEdited, stepFive);
// stepSix depends on stepFiveEdited, not stepSixEdited
addComplete(stepFiveEdited, stepSix);


This way, each additional step only takes one line instead of three, I think that this makes it a lot more readable (and also adding a new case is a lot easier). The other benefit is that it is easier to adapt this code if the specification changes. For example if the class name changes, or if instead of adding a class, the program should react differently, you only need to change it in one place.