The paint has to be of the correct consistency, not too thin or it will drip and run, but liquid enough to be rolled or dabbed evenly through the stencil holes.
Like every painterly craft stencilling has its own special brushes that come in different sizes. If you are using letter and number stencils in particular, you will find that it is so much easier to use stencil brushes than ordinary ones, because the long bristles of paint brushes push paint out under the edges of the stencils. However neatness isn't everything and Jasper Johns has worked with stencils of numbers and letters since the 1950s, using his orderly grids of stencilled forms to visually engage and play off against gestural brush strokes. In his case the paint sliding over and across stencil edges is part and parcel of the effect he is looking for.
The first time I remember using stencils to make images was back on my pre-diploma course, but I used a spray diffuser to apply the paint with, which I see are still available for only £1.95. We used them mainly to spray our charcoal drawings with fixative in those days and I have noticed that a very cheap low tech option now being used is to buy children's felt tip pens with a spray attachment. Back then I was very interested in Pop Art, and I used spray techniques to make paintings based on the images you found on seed packets. I'm sure I would have played around with felt tip pens and these children's spray attachments, if they had been available then.