What Licensing Should I Be Considering When I Take Open Source Software And Offer Up As An API?

I've done this a couple of times now. I took PhantomJS, and created my Screenshot API, and used ImageMagick to create my Image Manipulation API. These are two openly licensed software solutions, which I took, and am using as an API. 

What are my licensing considerations? If I keep my server side code licensed according to the specifications, am I fine? PhantomJS is licensed under BSD, and ImageMagick is Apache 2.0. Does the licensing extend itself to the commercial services I would potentially offer via an API interface? There are lots of questions to satisfy, before I move forward--I guess I am looking for a precedent.

I am evaluating at a number of openly licensed software solutions right now to deliver a variety of stops along the API life-cycle, ranging from design, to deployment, virtualization, and much more. As always, I am trying to better understand the layers involved, and how software licenses, patent, and potentially copyright might apply.

Just putting it out there to the universe, and curious to see what comes back.