Alternator...

Can someone please tell me what do I do with the "Green" wire now, do I just keep it open like that or do I look to ground it ?
Please, any help will be appreciated it as I seem to have a charging issue, the car starts and goes but at times if I try to start it too many times it dies out, but will always start if I just drive it for the day, park it and go back the next morning to leave with it again....its just that if I keep starting. shutting it off in a short period of time it seems to make the battery slow down and drained.
The strange part is the alternator was checked and it works fine ? I have about 13.7 volts (volt meter test while the car was running) but if the battery is removed the car dies, go figure. (At least Im trying still lol)
Thank you.
MAPS