What is the best way to make a puppet with huge red lips to look like lipstick?

Cut two pieces of red fabric in the shape of lips and stitch or glue onto the puppet. If you can find a shiny fabric then it would look like gloss lipstick. There are fabric paints you can use to paint the lips on.

BTW: Since you did not say I assume you are talking about on a hand in mouth puppets made of foam and fabric. On my marionettes I've simply painted the lips on then used high gloss clear finger nail polish over the color.