President Reagan has nominated Navy Cmdr. Michael J. Smith, who died in the explosion of the space shuttle Challenger, for posthumous promotion to the rank of captain, the Defense Department announced yesterday.

The promotion was expected to receive routine approval by the Senate. Smith, 40, was the pilot and one of seven crew members who died aboard the shuttle when it exploded Jan. 28.

When the promotion is approved, the monthly military benefits for Jane Smith, the astronaut's widow, will rise to $900 from the current level of $800. The monthly dependent benefits of $56 for each of the three Smith children will not change.

Mrs. Smith on Tuesday accepted an award from the Rotary Club in Winston-Salem, N.C., where her husband, a Rotarian and native of Beaufort, N.C., had been scheduled to speak.

"We miss him terribly," she said. ". . . But you have to be persistent. I still have to be a mother, and the children still have to go to school. We look to the future with the script Mike gave us."

Business leaders in Charlotte have set up a trust fund to pay for the education of the Smith children: Scott, a high school junior, Alison, 14, and Erin, 8.

"Contrary to what's widely believed, astronauts receive no special death benefits," said Bill Maready, an attorney who helped draft the trust agreement.