It sounds as if you are trying to proceed from the universal property of the enveloping algebra, rather than directly from its construction as a quotient of the tensor algebra. The latter may be a more helpful way to proceed for this particular question.
Here are some more precise hints:
(a) $U(\mathfrak{g})$ is a quotient of the tensor algebra by the relations $X \otimes Y - Y \otimes X = [X,Y]$. Think about how the ideal generated by these relations can intersect the degree one tensors. (The point is that these relations all involve a quadratic term.) If $\mathfrak g \neq 0$, is it possible for its image (thought of as the image of the degree one tensors) in the enveloping algebra to vanish?
(b) If you want to think in terms of the universal property, consider the adjoint representation of $\mathfrak g$ on itself. If $\mathfrak g$ is not abelian, this is non-trivial, so you get a non-trivial rep. of $\mathfrak g$, from which you should be able to draw a conclusion about the enveloping algebra. On the other hand, if $\mathfrak g$ is abelian, you should be able to compute the enveloping aglebra directly.