How do you get Articulate Storyline to publish faster?

I'm afraid that this is most likely going to be a hardware answer. It will depend a lot on the content you are publishing, for example video and audio can take a long time if you are using Storyline to compress the media. This can be speeded up by using another tool to compress your audio and video prior to bringing into Storyline, and ensuring SL does not do any compression when publishing. This can speed things up a lot, as you are compressing once, rather than every time you publish.

Again with video, if you can host it on a 3rd party service like YouTube or Vimeo rather than bundling in your course.

Keep your modules as small as possible. It gets very resource intensive when you start to have projects with a high number of slides.

Shut any other programs down that you are not currently using. 

Reboot if it is getting progressively slower. I think there are RAM leaks, that tend to build up over the day.

I have a high-end system (64gb ram, 10gb graphics card) but I will still see slow publishing speeds for content that has video and multiple slides.