Incorrect head alignment, dirty saving head, "dry" solder joints, oxide deposits on the head(s) over the years. Many things can have gone wrong over the years.
People usually prefer opening the Spectrum up to clean the heads, as many do not trust cleaning tapes. I would try to align and clean the tape deck heads as a first effort.
If that does not work out, it might be a (slightly) loose cable, and it will involve opening the computer and inspecting it.