Is MDPI a reputable Academic Publisher?

The peer review process generally differs between predatory and non predatory journals.

• Given that they have solicited a review from you is a positive sign.
• If you don't know the authors and the work before hand, then that is another positive sign.
• If the article you are reviewing is good, then that would be a third positive sign.
• Finally, if you make comments in your review and the authors address them in the published version, that would be a final positive sign.

If on the other hand you know the authors and work, it is crappy, and the comments are ignored, that would be a bad sign.

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.

In my very personal opinion MDPI is an reputable publisher, but their journals are not first-class. I did a review for them once, and the peer review process was smooth. Also, I knew the field of the reviewed paper very well, so I guess they selected me as a reviewer carefully. But I really can speak only for one of their journals.

On the other hand, I won't publish in their journals since they charge quite a bit for the open access, and there are other options in my field which are free for authors and subscribers. Also I don't like to be spammed with their "newsletters".

Added: I have recently seen this post reporting a problematic peer-review case with an MDPI journal. This lets me doubt, if my initial judgement was right. I suspect, it might depend on the journal.