Dishwasher rack is coated with nylon (https://en.wikipedia.org/wiki/Nylon)
Often that nylon coat is damaged mainly by methal objects (knifes, spoons, lids)
After the nylon coat is damaged rust starts to deteriorate rack and you end up with problems.
Rack is made from steel and steel tends to rust in contact with air / water. Nylon coat protect steel from rusting.
There is an easy fix for your rusty rack. First you need to remove rust , clean the surface and apply liquid nylon to problematic area. I have used special liquid nylon (http://www.plastidip.com/home_solutions/ReRACK)
Bought it from ebay for 10-15$.

Step 1: Remove Rusty Area

If rust damage is beyond repair the best way is to cut out rusty area.
You can use steel saw. Cut about 1 cm from rusty area where there is no
rust and there is no original nylon coat damage. Round up edges with sand paper or knife.

If rusty area is not that big and damage is reparable you can cut nylon from problematic
area then sand of rust and clean the surface. Prepare it for painting.

Be sure to remove all the rust! If you leave one rust spot, you will end up doing all the steps again…

Step 2: Clean Area With Alcohol and Paint

Clean area with alcohol. Use 80% alcohol if you can. Clean it twice.
Clean surrounding areas as much you can. Leave it to dry out.
First coat of Rerack apply thin. Leave it to dry. Apply second coat thicker
than first. Leave it to dry. Apply third coat thin and leave it to dry.
Leave 12 hours between coats. It need to dry well. It is best to over paint
area at least by 3 cm or more. On first paint attempt I have painted only
problematic area. After one-week nylon paint peeled off. It has passed over 2 months since this repair. Rerack is holding like first day. No signs of rust damage at all.