As a general rule of thumb candidates understand that employers will be receiving many many more applications for the position than they will be sending out, so it is understood that to write a detailed reply to every candidate would require an inordinate amount of time that could be better spent interviewing other candidates.
An additional point is that if people have sent out many applications and receive replies from them all it is much harder for them to read through it all, get their hopes out, find out its a cut and paste rejection letter, discard and repeat until they find the acceptance letter.
Remember the broader picture, you don't have time to waste sending out hundreds of rejection letters, they don't have time to receive letters of rejection from every position they apply to. It makes logical sense to avoid that step
So what do I do?
I see two alternatives here, you could simply say that you will only contact people on acceptance for interview within X days, that way the ones who do not get a reply know they have not been accepted. This saves you time and you only need to focus on the people you want to take it further with.
The other option is to email, phone, or inform somehow that they aren't being accepted, which as discussed above, helps no-one. It doesn't tell them why they weren't accepted, it won't help them in the future in any way. It certainly doesn't help you to halve your productivity to do so.
Personally I would say that you can do what you feel most comfortable with, there is no 'right answer' to some situations, there are just multiple approaches.
Is it worth the extra effort? That's a decision only you can make, and perhaps your company but you catch my drift.
Notice of Receipt
As pointed out by Bethlakshmi in the comments it would be advantageous for you to inform them that you have received their application. This can either be automated or manual but it's a nice touch that lets the applicant know that their application wasn't lost along the way.
I would say though that if you are going to pick an approach then make it uniform. Either reply to everyone or to no-one.
This let's everyone know what to expect from their application in an attempt to keep everyone on the same playing field.
Alternately you could write on your job listing that you will respond via letter to those who supply a cover letter, but this seems silly as a cover letter is not mandatory, it does not make one applicant any more experienced or prepared for the job than someone who did not supply one so there is no good reason I can think of as to why they should be treated differently.
If you really don't know. Ask.
If you really are unsure on what to do then there is no shame in speaking to your boss. At the end of the day you are public facing and are representing your company. If you are unsure what to do then ask others how they wish the company to be portrayed, and which approach would help you achieve that.