Parts of this piece are written for instructors/mentors and students, with the first part more directly for instructors. If you are just here for "what to do" advice, skip down to "advice I'm going to give students".
I've written before about how your grad school inquiry email better relate directly to the person you’re emailing and that a cover letter should center your expertise relevant to the position. But, I have now had numerous interactions that have compelled me to make a change in my scicomm course syllabus  this semester: adding in a module on professional communications.
My syllabus is packed (even though it's down to about half of what it was the first round). So, why am I adding in another layer? As far as I can tell, undergraduate students are not getting deliberate (or adequate) training regarding the importance and practicalities of writing professional emails. (Or most other dimensions of writing, which is why my co-author Stephen Heard and I wrote our upcoming book: Helping Students Write in the Sciences.)
Today, I'm also writing to say that I think you (as in we all, us faculty and instructors) should be teaching about professional emails.
Three reasons why teaching professional emails is our responsibility
As a first-generation college student myself, I had no idea what professional emails were supposed to be. (Granted, email hadn't been around long, especially for high schoolers when I was coming up.) I got lucky, though - I was raised a letter-writer, and I probably just went at email the way I did letters. I was also taught to type by following formulaic templates for office memos and the like. So, I got some aspects of professional emailing right, because emails got ported in from those analog genres.
For various reasons, the undergraduates we see today are in a similar boat regarding limited exposure to/models of professional email correspondence. I know this is true because nearly every email exchange I have with an undergraduate student makes this gap in training very clear. This has been the case at least as long as I've been teaching in higher ed (since 2015).
Someone has to to help these early career learners amass the professional skills they need to "cold-call" email people for internships, request new-to-them professors become their honors project supervisors, make effective inquiries about working in a lab or on a field team, and more. If I don't build it into my syllabus, these students probably won't ever receive this email advice. (I say that with confidence because (a) my course is a spring, senior-level course -- so, it's one of their last chances. And (b), I direct assessment for our department, so I can tell you: professional emailing is not part of the formal curriculum.) Of course, that means if you don't build it into your courses in some way, the students around you are just as likely adrift.
And a bonus fourth reason: the online advice is junk. I did a web search for advice about how to write a professional email, and let me tell you: if that's what our students have to resort to, it's no surprise we're not satisfied with the emails we receive. The materials I found were either were not comprehensive enough, not at all calibrated to an academic setting, and/or didn't offer a sufficient/accurate template.
So yes - I think it's our responsibility to share with students what we know about how people like us prefer to be contacted via emails. If you don't have capacity for that (and most of us don't, for all the reasons under the sun!), feel free to just link them over here.
I'm going to combine the advice below with an initial assignment, and then build email professionalism into the other major assignments. My intent is to provide students with multiple opportunities to practice and receive feedback during the semester.