In "Timescape", the USS Hera was lost with 800 hands. Backstage sources consider her Nebula class, thus identical in size to Galaxy, supporting the idea that no more than 800 people are needed - and many of those could be passengers, too. (Onscreen, we don't learn how big the Hera would have been, however.)
In the episode (actually "Interface") Geordi says about how the ship has "over three hundred people onboard". If the Hera
is Nebula-Class she seems very understaffed. However one explanation could be, that since they were just on a routine courier mission they were operating with just a skeleton crew.
We never saw her in uniform, so it's hard to tell. It's only in the novelization of "Emissary" that she is identified as a Starfleet officer.
I always assumed that Jennifer was a civilian. Her line in "Emissary" about watching out for junior officers seems an odd thing to say for someone who was in Starfleet. She is never seen in uniform and in the Mirror Universe on the Defiant
, Ben treats her like she doesn't know much about engineering/science (she has to remind him that over there she is Professor
Perhaps the Saratoga
was a survey ship, not intended for frontline combat and as such permitted to have family members onboard (she was pressed into service at Wolf 359 by being in the wrong place at the wrong time). Just a thought.