Preservation Hall, a French Quarter classic since 1961, is still flourishing today. The hall is open seven days a week from 8 to 11 PM, though I advise getting there by 7 to make sure you will be able to get in. The cover is very reasonable at $10 a person.
The lineup at Preservation Hall is always world class jazz, blues, or brass bands. The hall is small and intimate. The only places to sit are wooden benches, and there is no air conditioning, no amenities, and no sound system — but when the band starts, the frills don’t matter.