Mike — Instead of backpacking, you might try one of the car camps along the 38 and do day hikes outside the wilderness, e.g., the Santa Ana River Trail.
If you’re set on backpacking, consider Round Valley or Tamarack Valley, both easy hikes from the Palm Springs Tramway. You’d need to get a camping permit, $5 per person, available by mail or in person. I’d recommend calling the Mt. San Jacinto State Wilderness, 951-659-2607, to find out about availability of permits and of water. We hiked on the west side of St. Jacinto on Monday and saw some flow in some, (but not all), of the streams. You cam purchase your tickets for the tramway on-line – it saves waiting and also gives you a definite ride time.