Is knowledge better if it serves a purpose, like improving your art?

In my opinion knowledge that serves a purpose (improved art, medical science, applied engineering, take your pick) is better than "trivial" knowledge, but even trivia has purpose (it can entertain and inspire) and sometimes converts from trivia to "useful" knowledge when combined with new science and tech. A good example is pure math, here's a stackechange thread about mathematics areas that were found to have applications well after their discovery, by mostly mathematicians doing math for the sake of math.

Should art serve a purpose? Is beautiful enough, or should it be useful?

If you mean its physical form has a function other than to be perceived by humans for entertainment and inspiration, then sure, why not? But maybe not always. :) Also of note, not all art is necessarily beautiful. I would say a fair bit of "useful" or otherwise multipurpose art is quite ugly. Shock films, muckraking journalism, and hostile architecture come to mind as examples.