You're right that manpower depends on a number of factors, and global tensions were pretty high in the Navy of the mid-80s, so those increases could be entirely attributable to other more mundane reasons.

I can only find mentions of "+500%" Navy recruitment during that time, and stories of Navy recruiters setting up booths at movie theaters, so it seems like even the official story is probably somewhat sensationalized. I did find mentions of an increase in the total number of high school graduates entering the Navy, percentage wise. From sub-85% in 1985 to more than 90% in '86, so that could speak to the overall volume of recruits available during that time, but that might be drawing the wrong conclusions.

I didn't find any indication of similar things in the Air Force. As far as I can tell things were pretty steady. Either way, it doesn't really seem to follow that people would watch a movie about Navy pilots and then decide to hit up the AF recruiter.