You should consider the final output
Web vs Print are 2 vastly different mediums when it comes down to technical design specs.
RGB vs CMYK is the main difference which, when translating into actually printing something, will make or break a design. Additionally the final "deliverable" or output will change how effective and visible your design is.
Printers produce images using dot patterns to create prints and the human eye merges them into one color.
So if you have an extremely complicated pattern or have one with many overlapping elements/shapes you will have a hard time capturing the details. Along with complex details you need to take into consideration elements that have high coverage and common bounds.
The main consideration nowadays is responsiveness and device compatibility. Picking scalable and non-invasive patterns and designs that translate well to all mediums is key.
Focusing on visibility, spacing and ubiquity are the key components.
Simplicity is worth noting now too with the constant optimization of the web, you would want to have your art be able to load fast on nearly any device and also have fall backs if it fails to load.
Putting it all together
A design should be simple enough with few small or overlapping elements for printing. Even the best printers with the smallest picaliter print heads will have screening issues when dealing with complex or not significantly different areas; greys that are very close or muddy (4 color vs K only) will be a nuisance too.
Applying a design to a webpage you need to look at your final audience (web vs mobile) and how you want them to interact with your site. Visibility and UX is #1 always and forever. If your users can't easily navigate or view your site then you've failed.
So look to have highly contrasting, non-busy, patterns if combining or a simple non-intrusive base pattern if using just one.