All trails in the USFS San Jacinto Ranger District and the Mount San Jacinto State Park, including those served by the Palm Springs Aerial Tramway, are currently closed. The full recreation closures for both agencies are listed here.
The U.S. Forest Service issued an official forest closure order on the evening of 27th July, details of which are available at this link. This draconian measure initially closes the entire San Jacinto Ranger District trail system until 31st August 2018.
Hopefully common sense will prevail, and trails that are completely unaffected by (and literally miles away from) the Cranston Fire, such as Fuller Ridge Trail, Black Mountain Trail, Seven Pines Trail, and Marion Mountain Trail, among others, will be considered for reopening as soon as possible. Obviously these are functionally closed anyway at this time, given the combination of road closures (for non-residents), and the closure of the State Park at their higher elevations. Once those measures are lifted however, there is no logical reason to keep the aforementioned trails closed.
Bear in mind that this closure order applies only through August 31st. Some trails will remain closed long after that. Almost exactly five years on from the Mountain Fire, some trails are still closed from that fire (Caramba Trail, PCT from Spitler Peak Trail to Grethe Spring).
Fortunately, damage to high country trails by the Cranston Fire has been much more limited than by the Mountain Fire, but for example it is highly likely that the PCT south from Saddle Junction to Grethe Spring, the trail to Tahquitz Peak, and South Ridge Trail, will be closed well beyond August 31st. [I will be posting detailed information on trails impacted by this fire by the end of July.]
Exactly what the State Park does (and when) remains to be seen. It is not directly affected by the Cranston Fire at all, so hopefully it will reopen relatively soon. Of course many of the trails accessing the State Park initially pass through USFS land, complicating access (other than by the Tram) unless the agencies coordinate their approach.