The Dead Sea is just a one hour drive from Jerusalem.
Many people do day trips from Jerusalem to the area and try to cover all the sights in one day. In case you are short in time it might be a good choice, but if you want to really explore some of the places and get to chance to experience the quietness and atmosphere of the desert you can choose to spend a night or two in the Dead Sea area.
If it is summer, it will be too hot for camping in this area ( you just won't be able to sleep in your boiling tent) or in case you don't feel like camping, here are some options:
Right in the middle of the Dead Sea close to some famous hiking trails of the Judean desert. Not really a classic hostel, more of a camp atmosphere. Great view of the surrounding.
In case you are into something more luxurious after spending a couple of days in the desert sweating, climbing and getting sunburns, this is the place for you. located just next to the archeological site of Masada.
The hippie beach next to the checkpoint
All the info on how to get there here. Note that there are no facilities at all. Not even running water. you will have to bring it all with you.
The more touristic area of the Dead Sea. It is not allowed to camp on the main beach but if you'll set up a tent in the evening a bit far from the main touristic area nobody will have a problem with that.
I know people who found hosts in Kibbutz Ein-Gedi and Kibbutz Kalia.