If you check luggage you will likely have to collect it and recheck it. Since they are partner airlines, you could always ask at the check in desk.
If you hand carry it, you wouldn't have to go out and re enter security. But I would go to the KLM gate and check in. They should be able to do that there.
If there are issues, probably no help from Delta, especially since the next flight is on KLM (unless you booked it through Delta, then maybe.....) I'd call Delta and explain in advance and see what they can do. Better to be proactive than reactive.