Fractional Captain reported a terrain warning during approach. Approach was missed and subsequent landing was uneventful.

Narrative

Too low on a visual approach - We were en route to SGT. The airport didn't have weather available due to their AWOS was inop. I checked several of the nearby airports and they were all VFR. As we approached the airport from the southwest; still several miles out; the controller told us to call out the airport for the visual. We had asked for permission to make a turn to one of the fixes on the GPS approach to set up for the visual and it was approved. We were cleared down to 2000 ft; still several miles from the airport when I got the airport In sight and informed the controller. Once we were cleared for the visual approach; I told my first officer to descend to maintain VFR. There were some scattered clouds in front of us; so I didn't want to lose the airport.The mistake I made was I didn't specify what I altitude I wanted him to descend to.He turned off the auto pilot and started a rapid descent and soon after got a ground proximity warning sink rate and terrain."I was looking out the window keeping my eye on the airport and the terrain; I didn't realize how quickly he had descended; which was about 700 ft AGL.I immediately told him to climb back up to a safe altitude; which we did and entered a wide left base for runway 36.I flew through a rain shower; but I kept visual with the airport and the landing was uneventful.Suggestions: I should have specified an altitude for him to descend to. I we had a 10 hour overnight and a XA:55 show; along with dealing with a busy airport at ZZZ just prior. We had dealt with freezing rain and snow earlier in the morning; so all of this may have caused me to have been overloaded with work. Along with the weather being unavailable at our destination; I was under pressure to make it in on the visual.I should have waiting until closer to the airport to start the descent also."