The dye to use is NOT ink for silk screening -- It is for silk painting. Silk screening uses inks/paints that have a thick consistency and get pressed through a screen onto the surface of fabric. The idea here is to create a 'stain' as if dye bled onto the doll's face from red fabric that wasn't colorfast.  Also don't try permanent marker ink -- that is indeed immediately permanent and way too bright!

Another option is red fountain pen ink, diluted. It is quite a bit stronger and faster-acting than silk fabric dye; don't let it sit for too long without checking it the first time you try it.
 

• Good-quality liquid fabric dye for silk painting in red, magenta, garnet, etc. or mix  several colors to suit. Available at craft stores. It will look like a deep purple-black; dilute a few drops in water to see the actual shade it will come out as.  Linda O. was unable to find dye and used red fountain pen ink, diluted a little, and said that also worked well; I tried it and I agree but because it works faster I found it a bit harder to control.
• Hole punch
• Thick, durable paper towelling
• Fine-tipped tweezers
• Time (a week or possibly several weeks) and patience -- or  try Linda O.'s suggestion and use fountain pen ink and put the doll in the sun to dry. That seemed to accelerate the process so only a few applications, which dried quickly, were needed.

1. Use the hole punch to punch little round holes out of the paper towelling. Make several dozen.
2. Using the tweezers, dip two circles lightly in the dye, until they are saturated but not dripping. Blot them on a tissue if they have too much dye.
3. NOTE: Always use the tweezers to touch the dye unless you want to have pink fingers for a very long time.
4. Lay one dye-soaked paper towel circle on each cheek of your Betsy.
5. Pat the edges with the tweezers to make them lie flat over the curved  cheeks.  She will look like a clown.
6. Set her aside where she won't be disturbed. The first time, check after several hours to make sure the stain is not penetrating too fast or blotching. After only a few hours it should wipe off  totally with a damp tissue.
7. Leave the circles on for a day or two. They will dry in place. Peel them off and discard, wipe her cheeks clean, and inspect.
8. You should see a very faint blush, barely noticeable. Repeat dye application with 2 new circles until the desired color is achieved.

I have debated with myself whether this is an ethical alteration to make to a vintage doll. I have decided that if it lets someone enjoy and treasure an old doll who might otherwise be appreciated much less, then that's a good thing.