To determine if a book publisher is reputable, you'll want to do your homework. Start by researching the publisher online to get an idea of their reputation. Read testimonials from authors they have worked with, read book reviews, see if they have any industry awards, and take note of their pricing structure.
It's also important to make sure that the publisher provides the services and support you need. Ask yourself: Are they offering marketing, editorial, promotional, and distribution services? Can they help get your book into brick-and-mortar and online stores? Make sure the publisher is willing to answer all of your questions.
Finally, you'll want to make sure the publisher has a solid track record of successful book releases. Look for authors who've successfully published with them, as well as books that have gotten great reviews and good sales. If a publisher has proven authors and successful titles, that's usually a good sign that they are reputable.