When I say the word 'day,' I say /dɛi/, or perhaps /dɛj/. However, when I look at any dictionary that uses IPA, they always write the diphthong as /eɪ/. Why is this? Maybe my dialect of English (UK East Midlands) is just weird, but I feel like I hear it in other accents too? If I try to pronounce the diphthong /eɪ/, it sounds a bit like 'player', but without the pl and instead of a schwa, it's the vowel /ɪ/. Again, maybe my hearing is off, maybe my dialect is strange, or maybe I don't understand IPA well, but I feel like I'm missing something.
Whether it's pronounced [ɛi] or [ei] is just a matter of fact, and in my Midwestern American dialect, as a matter of fact, it's [ei]. And it's always [ei], so I don't know why anyone would choose a phonemic form that is any different, so that gives us /ei/ also.
For those who think they hear [ɛi], try comparing a hurried and very sloppy pronunciation of "Betty" and "bay". For me, after the [t] of [bɛti] is flapped, in a sufficiently sloppy style, that flap can be lost, giving a two syllable pronunciation [bɛi] and further the [i] can lose its syllabicity (which I leave implicit in my transcription), giving a contrast between [bɛi] and [bei].
Imagine that Betty from San Francisco is notorious for some reason; then "San Francisco Betty" would be [bɛi], clearly different from "San Francisco Bay", with [bei].
IPA vowels are based on Jones' cardinal vowels. In transcribing a word like "day", choices of letter are the result of numerous factors which are not articulated as rigorous rules of IPA usage. One pertains to the extent to which a transcription is phonetic, meaning that it strives to represent the pronunciation of a word, rather than the phonological analysis of the word. In that case, the choice of [ɪ] over [j,i] is based on phonetic similarity -- the second half of "day" is phonetically closer to the reference vowel [ɪ] than it is to [i].
This does assume a fixed standard for these symbols, which appears to be limited to some recordings by a very few experts. In the case of [i], the standard is supplemented by Jones' instructions regarding how to produce that cardinal vowel (the highest, frontest possible vowel whose constriction isn't so narrow as to induce turbulence). Of course, you are free to reject that standard, and many people don't actually employ a reference set of vowels in making judgements. Even if they do, auditory proximity is an art-judgment (it depends on a "feeling" by an individual), and is not a scientific measure.
As for phonological reasons to prefer one transcription over the other, there are vast numbers of mutually contradictory principles of some phonological theory that one can call on to make a decision. For example if you make it an axiom that English has no diphthongs (sequences of syllabic segments within a syllable), then any of [ei,eɪ, ɛi, ɛɪ] are excluded. If you make it an axiom that the glide [j] only appears in the onset, [ej, ɛj] are precluded. If you favor phoneme-minimization, you should prefer either [e: e] and dispense with [ɛ] entirely. There are very few phonological arguments that bear on the choice. However, there is a vowel alternation associated with contraction, so that the vowels in "she, they, I" become shorter and laxer before /r,l/ under contraction, thus [ʃi ~ ʃɪl, ðe ~ ðɛl, aj ~ ɑl] – unless in a particular dialect those are not the facts. From a phonological perspective, the choice is mainly driven by phonological ideology.