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What I currently do is save the following text on my computer, and simply paste it in an email response every time I receive this type of email.

I'm sorry to say that this email is too generic for me to consider you as a future PhD student working with me. Unfortunately, it doesn't demonstrate that you know what topics I work on. I'd recommend in the future sending only a few emails. Focus on max 10 or so potential supervisors that really fit your specific research interest. Explain what specifically drew you to them as a supervisor. Explain what you want to work on with them. You should have read one of their papers or at minimum read a description of their research from their webpage. The type of email you wrote will get very few responses because we receive several such emails. I hope this doesn't come off too harsh. I hope you might take this advice and improve next time if you send more emails in the future - because I really do want you to succeed.

"My name is ____________ and I am a senior majoring in Applied
Mathematics at UC Davis. I am interested in working in your lab during
graduate studies because of my broad interests in theoretical ecology.

Your website especially stood out because you state that you encourage
independent research and teamwork. Currently, I am collaborating with
six other students modelling plant population dynamics in fragmented
vernal pool landscapes. We defined our own research question, and
although it was the toughest part of the project it was the most
rewarding. During this project, I first came across your early work on
the population dynamics of plants with a dormant life stage - and
further reading some of your newer papers in epidemiology, predator
prey dynamics and evolution, I really like the diversity of the
problems in your lab. It seems like a really exciting lab to be part
of.

Specifically, I am interested in doing research in spatial ecology and
understanding how populations spread. This is because spread is not
only a fundamental process in ecology but also because it is at the
heart of controlling environmental pests and diseases. I think your
work on integral projection models will be especially relevant to the
spread models I want to study (integro-difference equations). I am in
the beginning phase of drafting an NSF graduate research fellowship
application on how to best choose the locations of pest control when a
pest species spreads according to such equations, and think you’d be
an ideal supervisor for the project.

I was wondering if you will have any openings for new graduate
students in your lab this coming year? Any comments or feedback would
be greatly appreciated."

cheers,

-Dr X

Note the parts about teamwork and independent research are specifically on this professors website that talks about qualities he looks for in graduate students. I didn't put this sentence in any of my emails to other professors (in fact all the sentences except for the first one are very different for each prospective professor).

The goal being to provide them useful advice that they might take and become more successful in the future. Often these students come from disadvantaged backgrounds or from developing countries and I think it is really important to try to help them and not just ignore their emails. Of course, as one becomes more well known, even this strategy might prove to be cumbersome.

Over this past graduate student application season, nearly all of the students I sent this email response to were gushing with gratitude for the advice.

If I am not especially interested in the student's profile, I simply delete the email. If no effort whatsoever was spent by the sender to address an email to me personally, I don't feel like I have a duty to respond to it.

If I am interested by the student's profile and they have just been clueless about how to get in touch with potential supervisors, you can answer generic advice as suggested in @WetlabStudent's answer, or you can write a one-line generic answer like "Thanks for reaching out. What would specifically interest you in working with me?" and give the student a chance to be more specific.

In our department, we forward those emails to the department administrator, who replies with a polite but generic e-mail encouraging prospective students to apply via our normal procedure. However, that’s a bit above and beyond, and I realize many departments don’t offer such a service to their faculty.

In lieu of that, if you are a Gmail user, I recommend the “canned responses” feature, which is even easier than the copy and paste approach IMO.

It seems like the consensus here about replying to generic emails with canned emails. I have nothing to add to the existing answers, except a technical solution to make the process of copy-pasting the emails easier. Two software I can recommend for Windows users are PhraseExpress and AutoHotkey. The former has a GUI and can sync with your phone, the latter gives you more flexibility.

For AutoHotKey, use:

:*:cemail::
SendRaw,
(
text text text
)
return

So every time you type cemail (shorted for "canned email"), text text text will be auto-typed. If you need to customize the text, use this template:

InputBox, string1, What to type?, Suggestion 1`, 2`, 3 
InputBox, string2, What to type?, Suggestion a`, b`, c 
SendInput text text %string1% text %string2%

The box will be like this:

Interesting question. I went through the application process a few years ago, and quite a few professors never responded, and to be honest I think that's perfectly okay, and have told many of my friends to expect no reply when applying. If you really want to, I'd keep it brief and say basically that you're looking for students with a clearer idea of their research topics and ones who know better what you are studying. and then tell them best of luck.

Life is about setting goals and priorities.

If your goal is to be a saint, then spend time helping these random unknown individuals to fulfil their ambitions. If not, just hit "delete".

Nobody can tell you whether to aspire to sainthood or to winning a Nobel Prize for your academic work (or something in between those extremes), but history suggests that trying achieve both simultaneously doesn't work.

How to politely respond to generic emails

Don't. If the email is unsolicited, you're not obliged to answer.

I know that this goes against our inner moral compass, but think of it this way: sending email is quick and easy. Too quick and too easy. Those people most likely have put less effort into sending such email than the effort you have put into reading it! It's you who's at disadvantage here. Responding is nothing short of letting yourself to be robbed of even more time against your will. That's why your colleagues just delete them. Don't feel bad for ignoring bad mails.

In fact, feel bad for replying! If you reply to bad mails, you're only showing that their strategy is working and encourage them to keep sending more. You're helping by ignoring.

You want to be helpful. But they're not asking for your help, they're asking for a specific favor. What you're currently doing is unsolicited advice, as unsolicited as their spam. And like most unsolicited things it only makes everything worse. You'll be seen as condescending and patronizing. You're basically saying "be more like me and less yourself". Nobody wants to hear that. You're not helping anyone, you just make yourself feel better about being yourself.

You should be careful to spend no more time replying to an email than was spent sending it. Don't feel bad about deleting an entirely generic email, but if I have time or the email has any merit I send a one-sentence email asking for details about how this relates to my own work, or if it is an undergraduate asking for an internship with a good email, I suggest in the future applying for a PhD with a proper connection to the work of the person they choose to contact. You can also write a web page about what you want written in PhD or postdoc "cold" applications (or saying that these are not welcome) and then just send the URL when you get these emails.