Here are a few arguments for print journals, some I’ve heard, and some I’d make personally.
Accessibility. Print journals can make journal articles accessible through the mail. This may seem an anachronism, but it still matters. There are locations in the world (including in developed countries) where the most convenient means of receipt is by post; rural areas with limited Internet access and no central library come to mind.
Interactivity. Print journals are still generally more interactive than their digital counterparts. You can hold them, make notes on them, record your impressions, clearly index them, and even cut out sections.
Curation of linked content. Each journal issue may provide multiple linked articles, which together tackle a question from multiple angles. This can facilitate better comprehension of a topic or encourage the reader to explore multiple facets of an idea. In the physical form, these relationships are made very clear.
Disconnecting. You don’t need to get on the Internet to read the journal; physical journals encourage you to break away from your computer and allow for a productive use of time in the physical world. You can keep it analog, if you will — somewhat of a luxury these days.
Anticipation. You can look forward to receiving a nice package of great articles with each issue. Rather than an article or two a day, you can get a nice meaty hit of intellectually stimulating goodness. It’s a great excuse to set aside a reading day, if your schedule is amenable.
Health. This may seem silly at first glance, but humor me here. Reading print is easier on the eyes for many people and can reduce eye strain. Further, you don’t continually stare at artificial blue light (a frequency in the visible light spectrum that is involved in the modulation of the hormone melatonin). If you plan on reading after sun down, a journal is probably a healthier option than a screen over the long term. There’s a lot of literature on PubMed, but here’s a nice overview.
Of course, those 56 reasons may not be particularly relevant to a faculty department — maybe 3 - 5, but probably not to the degree that they would make a print subscription worth it.
Specific to a faculty department? I’ve seen journals by water coolers and in reception areas, perhaps in the hope that they can encourage conversation or, simply, give visitors the opportunity to read about research that is relevant to the department domain(s) — not so dissimilar from academic posters that find their home in the department’sdepartments halls.