Backpacking trip in AZ

I called the Payson Ranger's Office and a recording said that campgrounds are open so I am not sure that the parking lot for Horton Creek would be closed.

When are you planning your trip for? Some ideas for a short backpacking trip near water:

East Clear Creek: start at the Horse Crossing Trail or the Kinder Crossing Trail and explore the Canyon. Anywhere downstream of the Kinder Crossing Trail is excellent. There is no official trail for most of the way, but the hiking is pretty easy so long as you don't mind getting your feet occasionally wet. See here: http://hikearizona.com/decoder.php?ZTN=426 for more info. I hiked this area just this last weeekend and the temps are starting to drop – lows were in the mid 30s. The trailhead (which can be hard to find/reach) is a 90-minute drive from Payson. PM me if you have any questions.

Parsons Trail: this is another great hike near water that is exactly 4 miles one-way. There are 4 or 5 river crossings and they are all ankle deep (except for the last one which is knee deep). Camping spots in the canyon are limited. See here: http://hikearizona.com/decoder.php?ZTN=86

East Baldy Trail: this is one of my favorite trails in AZ. It is a short and easy 4-mile hike beside the Little Colorado River to several excellent camping spots. This hike is in the White Mountains so the weather will be much cooler than near Payson. I would check the forecast in Greer for estimated temps. http://hikearizona.com/decoder.php?ZTN=2308

Edit: someone posted a photo-set of Horton Creek from Oct 13th (http://hikearizona.com/photoset.php?ID=27051) so the trail seems open. It is probably your best option if introducing kids to backpacking.

Maybe Chevelon Canyon — perennial stream, very pretty, and near Payson…

Thanks for the suggestions. I'm tempted to drive up there tomorrow to see what is open. When we call the Tonto Ranger Station the message said that everything was closed. Maybe the shutdown will end and we won't have to worry about it.