Based on the amount of spam that I receive from them (mostly calls for papers in special issues), the fact that their automated emails do not feature a “unsubscribe me” link, and the fact that I did not manage to be removed from their lists after several complaints, I would say that they are not a respectable publisher.
I don't know how long they have been around, but most of their journals in my field (chemistry) were very recently created: see there for a full list, which you can sort by journal creation date. Their older journal, Molecules, has a less-than-stellar impact factor of 2.4. Its editorial board, apart from the occasional celebrity or two, is unremarkable.
As a conclusion: I think they surf on the popularity of open access journals, but I don't consider them serious players.