How can I pay with both miles/points and cash for a flight on Delta?

Under the Delta Pay With Miles program, you can purchase Delta flights with a combination of miles and money, but you need to have one of the Delta co-branded American Express cards and purchase your tickets directly from Delta in order to do it. So, if you still have that Delta Gold AmEx, you can do this.

The eligible cards are the Delta SkyMiles Gold American Express, Delta SkyMiles Platinum American Express, and the Delta SkyMiles Reserve American Express.

Here's how you do it on their website:

1) Select your itinerary as usual with it set to show the prices in dollars, not in miles.

2) Select an option from the Pay With Miles drop-down box on the Trip Summary page (the page you see immediately after selecting your flights.) You will need to be logged into delta.com using the SkyMiles account that your Delta co-branded AmEx is tied to in order to see this option (otherwise, it will usually be replaced with an option to apply for one of the cards.)

Here's what the page looks like, with an arrow pointing to the Pay With Miles drop-down box:

3) Proceed with the remainder of the check-out as you normally would if paying for the ticket with cash only. If I remember correctly, you are required to purchase the ticket with the Delta co-branded AmEx if you choose the Pay With Miles option, but that might not be correct.

When using the Pay With Miles option, you are required to redeem at least 10,000 miles in increments of 5,000 miles. The redemption rate is 100 SkyMiles per U.S. Dollar of fare (i.e. $0.01 USD per SkyMile.) You can usually get better value than this by just purchasing an award ticket directly with miles, but sometimes this option can be useful if award ticket availability isn't good.