Should you as IT manager worry about the email writing skills of the other department employees? and what you should do?

I am not an IT manager, but I believe management is a process that you can apply to any.
As a manager I would be worried about anything my people would do that may cause the company sales or reputation.
Technology and the process is changing and improving rapidly.
As the manager, you should keep yourself abreast with all the new ideas in your industry, and train your staff to improve their performance.
Being worried alone is not acceptable in this brutal world.

Thanku Mohamed Rafie , this is a very interesting question ! Not only IT managers but the entire organization should be worried, infact very worried because Email is known to be by far the most effective customer relationship management tool.
direct marketing tool & healthiest form of direct marketing.
An organization gets in touch with its customer in different number of ways , n these instances of customer touch points are like moments of truth , its a moment to woo a customer n turn him into an evangelist from a mere customer.
This may generate huge number of opportunities to market our product, cross sell & up-sell & handle customer grievances.
So any mode of communication with customer must generate a feeling of delight.
Email writing skills are a vital component of any job an organization should invest in training & development of all the employees when it comes to communication with the customer or internally.

Dear Mr Rafi Do not worry too much.
You core skills in the field of your specialization are more important then this writing skill.
Just develop enough skill to put your answer in simple words.
People should get to the bottom of what you are conveying.
A customer will not be impressed with the Jugglery of words you throw at him.
Rather if people get confused then they start looking towards other people.

Actually not but giving some input to users may perhaps help them to use it effectively.
Perhaps take the initiative if not been taxed too much.
I feel very proud of the person who shares the knowledge and skills with others.

Of course yes.
I've been involved in training the users to, "How to write an email?".
It is critical for users to know that what they are writing and to whom? "KISS" rule is fantastic which stands for Keep it short and simple (Some say it Keep It Simple, Stupid).

If it is all about internal communications, then just leave it to them.
The very purpose of email is to communicate between two parties and if that is being met, then its fine.
But when it comes to the external world, your employee represents your company and thus their communication represents the company's outlook itself.
An awareness, a small training on the writing skills shall be provided to the concerned people if required.
Also an awareness on being GREEN to be done when the number of mails are too high.

Always worry as now a days Email is a major portion of communication.
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Better is to set by example him self and arrange small orientation and tranings...

simply, Yes

I don't see this issue as beeing an IT manager's responsibility. On the other hand e-mail is probably the most common communication media and I suggest that all staff must be trained to use it effectively.

I think he might help in awareness of writing and improve that buy conduct some sessions

In my opinion, the only effective way of managing and leading is by example. so i beleive its manditory for any Manager specially IT Manager to have very good skills in writing communication whether its email or traditional communication.