Yes, it is open year round. Here is a site with some details and the main excerpt:
“The first thing you need to know about camping in Joshua Tree is that the temperature here can change 40 degrees in any given 24-hour period. Yes, this is the desert, but the lows can be as extreme as the highs. Be sure to pack more layers than you think you’ll need.
Once the sun drops and you’re snuggled up inside your cozy bag, you’ll be treated to as brilliant a night sky as you’d find anywhere in the state. No overhanging trees to block your view; no city lights to wash it out. If there’s any park that was made for sleeping under the stars, this is it.
There are nine campgrounds in all, in addition to countless backcountry possibilities. Each of the campgrounds accepts RVs as well as tents, but generator restrictions minimize the trailer park effect. Even the busiest of these campgrounds benefits from being situated amid lots and lots of open space – escaping noisy neighbors is often as easy as taking a walk behind the nearest rock.
To reserve a site at the three campgrounds that accept reservations, call 800–365-CAMP or go to reservations.nps.gov.”
Wish I could go too!