For Noetherian schemes this follows from Serre's criterion for affineness by a filtration argument. 



Is there an example of a scheme X whose reduction X_red is affine but X is not affine?
No, if *X* is any algebraic space such that *X*\_red is an affine scheme, then *X* is an affine scheme. This follows from Chevalley's theorem. For *X* noetherian scheme/alg. space this theorem is in EGA/Knutson. As you noted, this can also be showed using Serre's criterion for affineness or by an even simpler argument (see EGA I 5.1.9, first edition).


For *X* **non-noetherian**, the following general version of Chevalley's theorem is proved in my paper "Noetherian approximation of algebraic spaces and stacks" (arXiv:0904.0227):


**Theorem**: Let *W*->*X* be an integral and surjective morphism of algebraic spaces. If *W* is an affine scheme, then so is *X*.


Recall that any finite morphism is integral, in particular *X*\_red -> *X*. As a corollary, it follows that under the same assumptions, if *W* is a scheme then so is *X*.


