Assuming your Delta flight is coming from the US, you will go through immigration/passport control for the Schengen area in Amsterdam. You will not need to go through immigration in Basel.
Per what you are saying, even though Delta and KLM are partners, and you could have booked as one itinerary, you chose not to and purchased two separate tickets, correct?
If so, then what you see for security, and how you proceed depends on a few things. If you will have checked luggage that you need to retrieve from the first flight, then you will go through immigration, on to baggage claim, then out into the arrivals are. You then will need to find recheck bags at KLM, proceed through Security, and onto your gate.
If you have only carry-on, then you can stay airside, follow the signs for connecting flights, you go through immigration for connecting flights, and onto your gate. Can't recall completely, but you may need a boarding pass to use the connecting flights path. You may be able to stop at a KLM desk in the international terminal to get one if you need it.