If this were asked 10 years ago, Tampa Bay easily. But not now, nowhere in the top 5, they've been relatively competitive and in playoff contention most of the last decade.
Detroit probably is the worst, though you can make an argument for the Cardinals. I give the Lions the edge as worst because unlike Arizona they haven't been cheap for the most part, just plain inefficient. The Saints are a close third, though much of the time it's actually looked like they were on the threshold of being contenders but something always impedes a breakthrough. The other two never seem to have any hope. I'd have to put the "new" Browns and Texans next even if they haven't been around long enough to truly qualify. Both teams have been abysmally run in the personnel department.
If you don't count the Browns and Texans, then I'd say the Jets and maybe the Chiefs. The Jets won a Super Bowl, but it was a long, long time ago. They've been somewhat competitive at times, though, but when they're bad, they're really bad, and bad for a while usually. The Chiefs might be a surprise, but outside of a few good playoffs teams recently, what really have they done over the last 40 years? They also won a Super Bowl a long, long time ago, but then were very mediocre for a long time. Even recently they haven't been in the playoffs THAT much.
I hesitate on nominating teams that have been playoff competitive a third of the time or more over the last 40 years. That lets teams like the Bengals off the hook, because they had a couple Super Bowl runs in the 80s and were good most of the 70s. Tampa Bay, Chicago, Seattle all fall close to that benchmark, too.