In my notary journal there is a section for placing a thumbprint from the signer. 

And I have the necessary requirements for taking a thumbprint: a thumbprint inker, and the knowledge of how to take a thumbprint.

But the question is: Should I require it?

First and foremost, I have to abide by the notary laws of my state. There are states that require a thumbprint. But Colorado notary law does not require a thumbprint for the notary journal. 

It may be a good idea to get a thumbprint, and there is nothing in the law that states that I can't ask for it. But I cannot make it a requirement.

Moreover, I cannot refuse a notarization if the signer is not willing to leave a thumbprint.

When performing a notarization, I have to stay within the boundaries of my role as a notary.  I cannot arbitrarily make up my own notary laws and requirements for notarizations.

Notaries should take their responsibilities seriously.  Just not too seriously.

There is a good book available for anyone interested in learning about how to take a thumbprint, why they are used, etc.

How to Take a Notary Journal Thumbprint: The How's and Why's of taking a signer's thumbprint in your notary journal (National Notary Association)