Destin is not the closest. But Gulf Shores (ALA) Orange Beach (ALA), Perdido Key (FLA), Pensacola Beach, and Ft. Walton Beach are closer and each one is spectacular. Panama City Beach is a bit gimmicky compared to the others. However, St. Andrews State Park and Shell Island is definitely worth a visit.
We'll be at Orange Beach for a couple of days next week. We chose it because it was close and looked fairly quiet.
But I've got a question for the OP - they said it was a "family trip" so I'm wondering if they want a beach with a big amusement park, etc...not sure if there is anything like that on the Gulf Coast?
The water clarity can change within a few hours or a few miles.