basically it's to avoid having to argue about space complexity, a practical detail which was theorized later. the point of the Turing machine was that one small set of rules about the actual logic (internal states of the machine) allowed the machine to compute everything which is computable. all that's needed is a lot of input (i.e. the program and data), a lot of space for output, and a lot of scratch space.
since no one had actually developed a universal computer at the time, details about how much tape would be needed would have been rather ambitious. so, "infinite" covered the bases. "unbounded" is slightly more accurate, and slightly less discomforting.
added: the theoretical question at that time (which, notably, preceded the common understanding of "programs" as we know them, and was really a metamathematical question related to, a la gödel, whether every mathematical truth had a finite proof) was whether it was possible to predict whether programs would be "never-ending". the answer, of course, is that it is not possible, but if the tape were finite, it would preclude asking the question.