I use a net with a good long handle and a large hoop. I am right-handed and hold the rod in my left hand. When the fish is alongside, I lift the rod so that my left hand is up near my left ear, to draw the fish as near as possible. Holding the net in my right hand, I tuck the end of the net handle in my right armpit, which provides good leverage to scoop the fish.
I only clear the one downrigger when I'm fighting a fish solo, so I keep trolling until the fish is alongside and ready to be netted. Then, I pop the boat in neutral, net the fish, and then put her back in gear. That way, I maintain control over the other side's gear and the fish.
I have the net mounted on a hook on the outboard side of the boat's cabin where it is easy to reach and will not get snagged on anything. I take it off the hook and put it beside me early in the fight, so that it is handy and I'm not reaching for it while the fish is in the danger zone at the side of the boat. Take a moment when you're laying it down to make sure it's not caught on anything.
When I mess things up (which I think is inevitable from time to time), it is usually because I haven't been careful enough to ensure the net isn't hooked on a cleat or something.