Making sure the under layer is completely dry before applying a stencil is an important step in doing it yourself, but finishing steps are equally important. Nail art will stay on for a time without anything over it, but a final layer or two of clear topcoat or a specialty fast-dry sealant will ensure that designs stay put much longer.
Some topcoats seal in color and dry in several minutes, and these may be especially good for stenciling because layered colors will be less likely to run or bleed into areas outside of the stencil work. Heated stations at nail salons make fast work of baking polishes onto nails in a very short time too.
If neither fast-drying clear coats nor salon tools are at your fingertips, it never hurts to kick back, relax and simply watch the paint dry.