An exchange visitor's spouse or child enters the U.S. in J-2 status and may stay for as long as the J-1 person is authorized to remain.
For traveling and re-entering the U.S., J-2 dependents need all the same documents as the J-1 would need as outlined above including a DS-2019 that is produced with the J-2's biographical information.
J-2 Dependents must have insurance that meets the U.S. State Department requirements during their entire stay in the U.S. While J-2 dependents have the option to apply for work permission on form I-765, the salary is not to support the J-1 exchange visitor. Since this process can be complicated, please see an international adviser prior to proceeding with work permission.