The transcriptions David Crystal has used for OP have so far been semi-phonetic. They were intended for use by actors, some of whom would not be familiar with the Internal Phonetic Alphabet, and so, to aid learning, a simplified transcription was used. Only those sounds in Early Modern English phonology that are different from Present-Day English are shown with special symbols.
The transcription was first devised in the context of British English Received Pronunciation, so the notion of ‘different from Present-Day English’ has to be interpreted accordingly when the transcriptions are read by people with different accent backgrounds. If you are a speaker with a post-vocalic /r/, for example, you would not need the special phonetic symbol for that sound. Paul Meier’s transcription of A Midsummer Night’s Dream, produced for an American company, illustrates some points of difference. But even within the British transcriptions there are some variations, partly due to texts being read in different styles and partly because – thanks to the experience of the various productions – our understanding of OP is now much greater than it was in 2004. The transcription notes used for some of the productions provide further detail.