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I see people close out emails and business letters with a comma after Thanks or Thank you.

Shouldn't a period come after Thanks or Thank you because they are elliptical sentences?

Instead of:

Mike,

Thank you for your letter of May 6 in which you said ....

Thanks,

Joe

Do this:

Mike,

Thank you for your letter of May 6 in which you said ....

Thanks.

Joe
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Hi,

Such things are just matters of convention. Neither way is unusual.

What seems a lot more awkward to me in your example is to say 'thanks' twice. I would either choose some other 'farewell' wording, or at least say 'Thanks again'.

Clive
Comments  

Further. Thanks.