I wonder what is meant by this question. In a very naive sense of the question, the type of mathematics Gödel worked on (foundational?) wasn't really computation oriented or even formulaic, so the ability to perform lots of calculations very quickly probably wasn't of much use to him. Indeed, computers where available to him in later years.

A more interesting interpretation of the question (to me) is what if he had access to computers used as mind amplification devices. For example; such as by using Mathmatica or Maple to explore and visualize theorems and things. I'd imagine the benefit of this activity for someone like Gödel would be "not much". Computations and simulation have inherit limitations; precision and rounding errors for scientific computation, and the fact they can only model what we can imagine for another. These people such as Gödel, Neumann, and their ilk, new this, they begat the era of computation we have today, and all the limitations that involved. Neumann in particular was famous for, when presented with your problem, would tell how to solve it.

What's new today that they may not have foreseen is the vast level of internetworking and human communication that arose from ubiquitous presence of computers and networks.

Something that strikes me about the present article is the fact that Gödel, keen to see Rucker before he knew him, was not so keen to converse with him after their first encounter. One might think Gödel was not too impressed with Rucker, maybe found him boring and dull for example.