There are explicit photographs posted outside the buildings of some of the sleazier clubs on Bourbon St. Often you will also see young (er-r-r) "ladies," scantily- clad, standing in the doorways of these places. They are not particularly aggressive, even towards potential customers, so I doubt if they would personally bother you or your children. Most of the heavy crowds and general nastiness of Bourbon St centers around evenings and weekends, but be alert at all times.
My suggestion to you, given the financial arrangements you've already made, is to just go with it, but to exercise caution (for both safety, and the sake of innocence) by keeping the kids with you at all times, and by promptly ducking down the side streets, from Bourbon to Royal, at every opportunity. Perhaps your Hotel has side entrances so that Bourbon St can be avoided entirely? Or, if you are really uncomfortable, have a cab pick you up and drop you off at the front door.
The idea of children on Bourbon St is not unheard of, but I think we're all wondering how you managed to book a trip to New Orleans with "no clue what Bourbon St was about."