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I am working at a company which sends millions of SMS for all sorts of customers. Many banks are among them. For some customers, we have stored message templates in our database. Every once in a while, the customer contacts us and asks to modify some of their templates. These templates come in several languages. As one of them is my mother tongue, I often realise that there are typographic or grammatical mistakes in those texts.

At my company, I am currently the only one who cares for those messages whenever the customer asks for a modification. Although it is none of my business, I feel that I may tell them.

So, my question is: Should I address this at all? If, what is the best way to address it via e-mail?

As you mention there is no urgency and you have a direct contact with the sender, I would suggest coming back to them with something like that :

Dear client,

I've received your request and implemented the changes in our
database.

However, It's come to my attention that in the template I
received, you wrote the word 'FouBare'. Maybe you meant to write
'FooBar' ?

Please tell me if you would like to make modifications or
publish the message as-is.

If you do so, you'd have to be sure you are correct, and that the client is not in fact using obscure or technical vocabulary that would be grammatically correct nonetheless.

Note :

Even if you see common typos or very elementary mistakes, I would always come back to the client with the suggested corrections before putting them in the database. That way, you have an email trail and the client has to directly approve the final modifications.

Should I address this at all? If, what is the best way to tell it via
e-mail?

Quality is everyone's responsibility. So yes, you should try to find a way to address the quality issues you see. And preventing an error is almost always preferred over discovering an issue in production.

Talk to your manager, or the project manager, or whoever is responsible for owning the client relationship.

Mention that you occasionally see issues in some of the templates, and ask how you could best convey the issues you are seeing so that they are corrected before being sent.

In some companies, a bug report would be appropriate. In other companies, a quick note to your company's client support person would be appropriate. In a few companies, they would want you to respond directly to the client.

Just ignoring an obviously incorrect template isn't helping anyone.

Where I work, our systems are responsible for producing printed output and emails.

I speak both English and French (both of Canada's national languages). When we see an error we are instructed to leave it alone. The client is supposed have all of their content reviewed and translated appropriately. The actual content itself is not our concern.

When you start doing these types of services for clients, they come to expect it and before you know it, you will be the final line of defense against spelling mistakes. Not to mention, if anything goes wrong with your interpretation of the correction, your company will be the one on the hook.

The reason we are told to leave it alone is because if we recommend a change, we become liable for any fall out that is causes. Feel free to tell your team leader, but that's as high as I would go with it. If your team leader wants to assume the risk, or welcome the extra work, that's on them.

Do your company charge for text verification?

If not then it's not your problem and not you responsibility and you would be doing somebody else job for free (for example the translation agency that get paid for that)
If yes then it's the corrector job.

You can just let your company know that there is a profit to be made by offering text check for customers.