The Frenchmen Hotel is a must for anyone keen on exploring the cultural scene in New Orleans, without any of the trashy chaos of Bourbon Street. We also found it extremely affordable for what you're getting out of it.
If you're looking for a spot at the edge of the French quarter, in walking distance to virtually everything, with all the jazz, blues, deep roots, and southern cuisine that Frenchmen Street has to offer, then this is the place for you. There's a gorgeous pool under an open sky available to patrons at all times, the staff are very attentive and kind, and the rooms are fun and clean. The only thing to be aware of (and they'll be the first to tell you via text message in advance of your stay), it is very loud right up until around 12am for most nights, and up to 3am on Friday and Saturday night. But if you're willing to plan around that rather than resist it, you'll be fine! You're on vacation in NOLA for goodness sakes! Live a little :)