Believe it or not, but drips are actually an intentional strategy in painting. Drips have a loaded history - directly references Jackson Pollock when he shocked the world with his absolute abstract expressionism in the early 1950's. Today in contemporary art, "drips" are actually a "thing of the past" so to speak - most often referencing this type of painting and this era of "Abstract Expressionism" so it might be helpful to be aware of the context of using them but it's not necessary obviously. Only in the academic circles does this stuff get analyzed to grainier level!
But I will add that when I was at the Painting's Edge Idyllwild Residency lead by Roland Reiss in the summer of 2008, I had a critique with the famous Pat Steir. One of my paintings that we were looking at in my portfolio I had turned "sideways" so that the drips were going "horizontal." She actually looked straight at me (she was well over 75 at the time - so she had been the generation following Pollock, etc.) and said "Drips are gravity, you should never have the drips go sideways that's just not done. You should rotate so the drips go "down." Important stuff! :)
In this painting above, the drips have become an integral part of the overall aesthetic. It looks almost like "stems" from some floral arrangement. I haven't finished this painting yet - still in progress but so far, I plan to keep most of the "drips" towards the bottom of the painting. With this painting, it also is practical. The painting is pretty large (4 feet x 6 feet) and it is on panel so it's actually difficult to put on the floor so this easy and most practical thing is to keep it leaning against the wall to paint. So in this way, the paint drips are inevitable. But I haven't worked with drips in a while, so I think the effect is lovely. The drips here are very straight-forward, used only with water. You can use matte mediums and gels to get "slower" drips that build up from the surface too.
These drips in this painting are starting to look very much like series of "line" and the line quality, even though random, is really quite lovely as well. But who knows what the finished will look like. This is another example of practicality - this piece is a large paper piece and there is no way right now in my studio, that I can put any of it on the floor. But I LOVE the bottom - probably will try to keep most of it, the random drips, drops and lines that are all completely accidental!