Welcome, my name is XXXXX XXXXX I will do my best to help you with your issue.
Where do the down spouts empty now?
So let me see if I have the current picture. You have new downspouts that dump on the ground and you have the old buried drains stubbed up out of the ground with nothing connected. Where are the new down spouts in relation to the old ingroung drains?
Ok, it sounds like the downspouts should have been connected to the existing in ground drains. What I suspect is happening it that the water from the down spouts is not draining away from the house like it should. If the grade doesn't slope away from the house and/or the downspouts can't be extended so the water drains away from the house then the downspouts must be connected to the in ground drains. I assume water in the basement was not a problem prior to the new gutter installation
If the old system was connected to the in ground drains and you had no water in the basement then capping the in ground drains is not going to solve this problem
Ok, I'll tell you how to plug the in ground drains but it may not solve your problem. You can plug the drain openings with an expanding test plug. This would be my first choice since it will allow you to remove the plug if you find you need to use the drains in the future. You can get one at any plumbing supply house. Be sure you get a plastic one