As a general rule, you should join the mileage program for the airline that you want to fly when you redeem your miles. While it is possible to redeem miles on partner airlines, there are several disadvantages:
- Availability is typically much more limited when trying redeem miles on partner flights. Also, it's often impossible to check availability on partner airlines using the online site and you will have to have an agent assist you.
- Mileage "prices" are not always symmetrical. It is sometimes more expensive to reserve a reward flight on a partner airline than it would be to use miles on the same airline. For example, reserving a Delta flight using Korean miles might be more expensive than booking the same flight using Delta miles.
- Frequent flyer programs are usually targeted to the airlines home-market. Mileage credit card offers and other partner promotions are typically targeted to those customers. For example, credit cards that are only issued in Korea, hotel partners only in Korea, etc.
- Finally, it may be more complicated to book reward tickets outside of the home-market (no local phone number, website that takes foreign credit cards etc.)
That being said, in some cases when it is possible to earn miles faster on the "other" airline, you could consider using their program. For example, if you have an elite status, use a mileage credit card or the airline is offering other promotions. In this case, you should consider the trade-off of earning miles faster versus the problems mentioned above.
EDIT Just noticed, for example, the Delta offers a special "Low" economy class reward for only 12,500 miles whereas the cheapest reward using Korea miles for a USA domestic flight is 25,000.