Remember: syllables are nothing to do with spelling.
If you want a simple, everyday rule of thumb to "how many syllables in this utterance?", then you can generally equate this with "how many beats the utterance would be clapped to". So from that simple perspective, "cry" would be one syllable.
From a theoretical perspective, things are obviously a little more complicated. You can see a syllable from two main angles:
- a structural angle, in which you look at what clusters/combinations of sounds can generally occur in a given language and then derive that language's underlying syllable structure from those combinations;
- a phonetic (or at least, broader phonological) angle, in which a syllable corresponds to a peak in "sonority" or "vowelness" (or some such notion).
- even from a theoretical perspective, even intuitive notions such as "how many beats" an utterance has or "which parts of the utterance would be aligned with strong beats when it is put to song" can also constitute evidence.
Usually, even from this more detailed theoretical perspective, the word "cry" would still regarded as having one syllable. The diphthong /ai/ then forms the nucleus of the syllable, and the cluster /kr/ the onset.
Why? Why not say that in /krai/, there are 'two vowels', a /a/ and a /i/, and hence two syllables? Well, a mixture of evidence such as the fact that the /ai/ appears to be 'pronounced as a glide' (as clearly opposed to, say, in "Aïda"), the fact that speakers would clap it to one beat, the fact that in the process of 'changing one word into another' in English, there clearly seems to be a common case where /ai/ can be substituted as a unit for another vowel (e.g. "creed" ~ "cried", "hide" ~ "hid"), and intuitively speakers wouldn't see this as 'more of a change' than, say, the change from "hide" to "how'd". Or put another way: if you change /ai/ to /au/, intuitively to a speaker, it doesn't feel like you've 'only changed one of two vowel sounds', but rather it feels like you've simply 'changed the vowel').
There are more problematic cases, however. For example, processes of phonetic assimilation can make what we might intuitively think of as 2 syllables in practice be merged phonetically into a single syllable. Certain consonants, notably /s/, can also correspond to peaks in sonority-- there's even an argument that at some level, a /s/ after a complex cluster (e.g. "strengths", "widths") actually corresponds to a separate syllable, so that e.g. "widths" would consist of 2 syllables.
The number of syllables in "cry" is usually less controversial, though.