Remove the door handle.
With angle grinder and wire brush attachment remove the rust.
De-grease the area with POR15 Marine clean.
Kill the rust with POR-15 Metal Ready / KBS rust blast.
Apply 2 coats of POR-15 rust stopping paint
POR15, Inc. - Stop Rust Permanently - Repair Gas & Fuel Tanks
KBS Coatings - Paint Over Rust - Stop Rust Paint - Gas Tank Sealer - Gas Tank Liner
This will stop the rust and stop it from spreading any further. It is likely that you'll have to treat both sides of the door.
The only problem you have now is that you'll have a black patch on your door.
To get it back to the original colour you can use POR-15 self-etch primer (also available in an aerosol) and then normal products such as a high build primer and then your colour coat (code for this colour can be de-coded by the dealer and perhaps some of us here if you provide your VIN) and possibly a clear top coat depending on how your colour coat is made...
In short a lot of work to do it properly.
Please note I have no shares in POR-15! Honest!
I have tried their products as well as the KBS ones and I can't tell the difference between them.
Others recommend Eastwoods products
Auto Body Repair Tools Shop, Auto Body Supplies, Automotive & Classic Car Restoration - Eastwood
To date after buying loads of their stuff I have only had trouble with one of their products and to be fair I was using it incorrectly - so Eastwood's stuff also good.