I agree - the 'primary' condensate drain appears to be clogged.
It appears you are seeing water coming out of the secondary condensate drain - you will only see water coming out of there if the primary is clogged.
Your secondary condensate drain should probably be connected to ~something~. Either a secondary drain line or a float switch that will cut off your AC when the primary gets clogged.
DON'T plug it with putty - if you want to plug it, you should be able to get a threaded nipple from the hardware store for $0.39. If you do plug it, be aware that it will still cause flooding if the primary drain is clogged - the water will drip down inside the unit to the floor where it will be captured by the drain pan that your furnace is mounted on... Your furnace does have a drain pan, doesn't it??
If you can, please post a photo of your AC unit and condensate lines. It will help the forum members to make recommendations.
Here is a good article on the subject:
(Note: The other DIY Chatroom members that recently educated me in the ways of the condensate - you'll probably see some additional posts from them shortly.)