Do you mean crypts as in places for burial, or crypts in the sense of vaulted basements of the church?

Access to monastic churches was restricted, though the exact details varied from site to site and from order to order. Some orders, such as the Cistercians, officially completely banned the laity from using their churches (though there were exceptions, for instance at Hailes), others allowed the laity to worship in the nave (the nave might even be the parish church) but kept them out of the rest. The choir and the liturgical areas around the high altar were always private. Where the church had a shrine that was the object of pilgrimages the shrine was usually located in the east end of the church beyond the choir and pilgrims would have access to the nave and the shrine area via the aisles and transepts, often laid out as a processional route. If the shrine was in a crypt, as at Worcester Cathedral Priory, then pilgrims would have access to it, otherwise not.

Founders, patrons (usually the descendants of the founder) and major donors expected access to the church, and often burial rights so as to use the monastic church as a family mausoleum, it was pretty much an expected part of the deal, and while some orders tried to discourage it (i.e. Cistercians, once again, and Carthusians) they had to play along for both political and financial reasons. Other orders encouraged lay burials in their churches, not only could they charge for it, but it also encouraged people to donate to the place where their friends and family lay and helped to cement alliances with powerful local families.

Regarding dignitaries, if an important neighbour, a powerful lord or clergyman or even the king asked for a look around the church or crypt they'd probably let them whatever the rules said so as not to risk alienating them, and to demonstrate the wealth. power and holiness of their institution. Ordinary folks, not so much.