Unless you're going in the next week or so, I would say go up the middle. Some of the roads around Katherine are closed and I doubt things would be helped with TC Craig in the area.
If you do go up the middle, definitely spend more than one night in Alice, and make sure u stop in at the Daly Waters Pub (about 250km south of Katherine i think) - we had a very late start from there the next morning
. Also check out Mataranka Hot Springs about 100km south of Katherine.
The backpackers in Broome are looking good as I type this
Cheers,
TL