How safe is it to provide my credit card information to an admissions office? Recently, I got admission to a PhD program in the United States. And they have prepared my I-20 and the other documents to send me by mail.

They have offered me to provide them my credit card number and expiration date so they can send the documents by express service. They also offered a free regular service too.

How safe is it to provide my card information to an admission team? I know, probably, they will not misuse it, but do they protect it afterward too?
Do they have any obligations for the protection of the information. 

Is it a common offer in all the USA?
I wouldn't send any personal information like that through an email, even if I were sure the email was sent from a secure and trusted location. Whether this be for an admissions office or anywhere else.

My advice would be, if you wanted it, to call the admissions office directly and provide the information over the phone to get the express service. There is an added fail-safe of you looking up the number and calling it yourself to ensure it is legitimate. 

Otherwise, just take the free postage if you are that concerned about having your details stolen.