How should I safeguard my FAA certificate number? The FAA’s Search Airmen Certificate Information reads in part

Personal Information
We do not display your SSN, certificate number, or date of birth. We will display a releasable address in accordance with the Aviation Investment and Reform Act for the 21st Century (PDF). We display medical information based on FAA’s Civil Aerospace Medical Institute (CAMI) medical records. If the medical dates are incorrect, please contact CAMI at (405) 954-4821.

Airmen Certification Frequently Asked Questions contains the same intended exclusions.

Do you display my certificate number on the Internet?
We do not display your SSN, certificate number, or date of birth on the FAA website.

The U.S. Department of Transportation publishes an FAA Privacy Impact Assessment – Airmen/Aircraft Registry Modernization System (RMS) notes that Social Security Numbers were used in the past for certificate numbers, but policy has changed.

Personally Identifiable Information (PII) and RMS
To handle airmen certifications and aircraft registrations, FAA requires PII about airmen and aircraft owners/registrants. With this in mind, RMS may include or collect the following data on airmen and/or aircraft owners and registrants, either for authentication or certification requirements: Name, Date of birth (airmen only), Social security number (airmen only), Driver’s license number, passport number, or government ID number, Physical Description (height, weight, hair and eye color, sex, and citizenship), Address (airmen only), Medical records (airmen only), Certificate number (airmen only). In the past, RMS used an airman’s social security number as the certificate number. FAA is now changing that practice, developing unique certificate numbers not affiliated with social security number. During this change, airmen can request, online or offline, that his or her social security number not appear as a certificate number. For an individual’s PII to be included in RMS, that individual must be associated with an airmen or aircraft certificate or application.

It is clear that for airmen whose certificate numbers match their SSNs, adhering to best practices and remaining clear of practices to avoid (cf. recommendations from the Social Security Administration) are wise courses of action, primarily to reduce the risk of identity theft and fraud.
To get credit for FAA WINGS training hosted at AOPA’s Air Safety Institute, certificate number is a required field—but also one whose value ASI is careful to note that they do not store and that they pass with some level of encryption protection.
So many legal disclaimers raise red flags. What are the risks of disclosing an FAA airman certificate number? Is all the handwringing due strictly to the fact that some certificate numbers are also Social Security Numbers and nothing else? How should I protect my certificate number, which differs from my SSN?

Note: All quoted passages were retrieved from the linked resources on July 26, 2015.
I guess the question is, what can someone do maliciously if they have your certificate number, and what's the risk of that happening?

My assumption is that they can do nothing - and therefore the risk is zero - because no one uses the number for anything: it isn't needed for medicals, training, TSA approvals, flight reviews, checkrides, renting aircraft, buying insurance etc. People do want to see your certificate (and even take a copy) for various reasons, but no one will accept your certificate number by itself as proof of anything. That's my experience in the part 91 GA world anyway; I have no idea if certificate numbers are more important for airline pilots.

So my guess - and it is a guess - is that this does indeed come from when SSNs were used as certificate numbers. And as a result, unless your certificate number is also your SSN, there's probably no reason to do anything special to try to protect it.