The incident occurred at 8:30 a.m., the base said in a Facebook post.
"Local law enforcement and first responders are on the scene. First responders have extinguished the fire and are securing the area. The pilots have been transported to a local hospital for evaluation," officials said, adding there was no immediate threat to the area.
The incident marks the second crash of a T-38 trainer in six months.
- Air Force Identifies Pilot Killed in T-38 Crash
- Air Force Wants to Predict Aviation Accidents Before They Happen
- Amid Mishap Spike, Air Force Orders One-Day Pause for Safety Review
Barbour, a native of Van Nuys, California, was the aircrew flight equipment flight commander with the 47th Operations Support Squadron and an instructor pilot with the 87th Flying Training Squadron.
Capt. Joshua Hammervold, an instructor pilot for the 87th FTS, was injured in the accident.
The Air Force has lost 18 service members since November to aviation mishaps.
As of May 2, manned aviation Class-A mishaps -- defined as involving fatalities, severe damage totaling $2 million or more, or a complete loss of the aircraft -- have increased 48 percent in fiscal 2018, officials said recently.
The latest accident comes as the service is conducting a staggered one-day stand-down ordered by Chief of Staff Gen. David Goldfein on May 7 to give units "the chance to identify issues that they can work and elevate up to the [major command level] ... and the Air Staff if necessary," said Maj. Gen. John T. Rauch, chief of safety for the service and commander of the Air Force Safety Center.