How Important are E-mail Sign-Offs?

Very! Those you communicate with will be looking at your e-mail in its entirety to determine your overall tone. For example, if you send a rude e-mail and sign-off with "Have a nice day!" then it is clear that your that you intended to be sarcastic and the receipient will learn a bit about the type of person you are. Your sign-off can portray the overall demeanor behind your e-mail.

Most folks stick with the standard:

Best,
Regards,
Sincerely,

But your sign-off need not be standard! Nor does your sign-off have to be a word or phrase above your name to add to your e-mail's overall impression. For example:

HTH! (Hope this helps!)
Have a great day!
Thank you so much!
Talk to you soon! (TTYS)

Your sign-off gives you the opportunity to ensure that the sentiment of your e-mail is clear and not open to interpretation or the recipient's speculation.