Generally include mirror links, but don’t use them “as a crutch”
If you are designing your email (instead of just typing it) and you don’t know who is receiving it, a mirror link is generally a good thing.
“The Browser Wars” never came to email clients, and capabilities of email clients vary greatly. And testing is more difficult for emails, since you have the added dimension of the web client (e.g., gmail.com) which can run on various browsers.
It’s good for your brand, in general, not to send users things they cannot read, even if they are a small minority or users.
If you don’t have a good reason not to have a mirror link, include one.
Exception: platform-filtered email lists
If you control your email list very tightly, you can sometimes skip the mirror link. For example, if you have an email list that is limited to users who read their emails exclusively on iPhone, you might develop to that platform and skip the mirror link.
Your conversion analytics will tell you if that is a good strategy for you.
Vital and personalized data
For emails that contain vital information (instead of sales material) it is typically best to under-design them, so there is a high likelihood they will simply work. This is especially true for transactional emails that are not sent at a time of your choosing.
Emails that contain personal data would need to be behind a login wall on your website. Therefore, it is doubly important that your user be able to read the email rather than relying on a mirror link.
Personalized, but not personal content
If your email campaign involves sending recipients personalized content, it may not be appropriate to have that content available at a public URL.
Of course you would not want identifying information in the publicly-available URL (a la, Dear John Smith, we thought you’d like the following embarrassing products). But even if user privacy is protected, such a system could reveal competitive intelligence to competitors.