All airlines that participate in the Civil Reserve Air Fleet (CRAF) program do so through contracts with the Department of Defense (DoD). These contracts define what the airlines will provide if the DoD activates the program, and the compensation the DoD will provide the airline both during the activation and during times of peace. There have been many such contracts over the decades and perhaps some of them offered specific personnel for the DoD to call upon, however the basic style of these contracts is just that the DoD is able to request some number of aircraft to perform an airlift service.
In short, the DoD is essentially calling upon the airlines to operate flights at certain locations and times, but the actual specifics of performing those flights are left to the airlines. Although providing the personnel needed to operate the aircraft seems to be part of a CRAF contract, the airlines themselves determine how they assign their personnel to a CRAF trip.
The Code of Federal Regulations (CFR) codifies the various rules and regulations that the US government has to follow, and Title 32 of the Code of Federal Regulations (32 CFR) covers national defense topics, such as the CRAF program. Part 243 of 32 CFR lays out the guidelines for how to determine compensation for CRAF contracts, and 32 CFR § 243.3 is a section of helpful definitions about the program:
Civil Reserve Air Fleet International Airlift Services. Those services provided in support of the Civil Reserve Air Fleet contract, whereby contractors provide personnel, training, supervision, equipment, facilities, supplies and any items and services necessary to perform international long-range and short-range airlift services during peacetime and during CRAF activation in support of the Department of Defense (DoD).
As used herein, CRAF aircraft are those allocated aircraft, which the carrier owning or otherwise controlling them, has contractually committed to the DoD, under stated conditions, to meet varying emergency needs for civil airlift augmentation of the military airlift capability. The contractual commitment of the aircraft includes the supporting resources required to provide the contract airlift. In return for a commitment to the CRAF program, airlines are afforded access to day-to-day business under various DoD contracts.
There are a variety of other rules, regulations, and laws covering how the contracts are negotiated and what the contracts can contain, however the basic style of the contracts are clear: the airlines offer the DoD some number of aircraft to call upon to provide airlift services, and it is up to the airlines to provide the equipment and crew necessary to actually perform those services.
I'm sure each airline has their own method of assigning crews to pledged aircraft, and they likely run the full spectrum between 'entirely volunteer' to 'part of your employment contract'. However, just to have at least one example I was able find the CRAF procedures for United Airlines pretty easily. This document appears to be an agreement between United and the Air Line Pilots Association (ALPA), and Letter of Agreement 20 on page 413 of the document (423 of the PDF) has this to say about fulfilling CRAF requirements:
2-D-1 Given the emergency nature of the CRAF Operation, the Company shall keep on file a currently effective Preference List which shall include, in order of seniority, all pilots desiring to fly the CRAF Operation. Each Pilot on the CRAF Preference List shall indicate whether he is preferencing a Captain vacancy or a First Officer vacancy, and Equipment type.
2-D-13-c CRAF Trip Assignments
Whenever the company assigns individual CRAF Trips to pilots (when the assignment
window opens) it shall do so in the following order:
2-D-13-c-(1) To a Reserve who is available for the assignment and has volunteered to perform CRAF flying,
2-D-13-c-(2) To a Reserve who is on a day off (and who is willing to move his day off) and has volunteered to perform CRAF flying,
2-D-13-c-(3) To a Lineholder who has volunteered to perform CRAF flying (and who is
willing to trade his scheduled Trip for a CRAF Trip),
2-D-13-c-(4) To a Reserve who has not volunteered to perform CRAF flying who shall
be advised that the Trip is a CRAF assignment. In this event the Reserve may refuse
the assignment due to the military nature of the Trip and shall be bypassed for that
assignment with no indication,
2-D-13-c-(5) To an out-of-Base Pilot who has volunteered to perform CRAF,
2-D-13-c-(6) To a Pilot in accordance with the provisions of Section 20-H-5 who has
volunteered to perform CRAF flying
As you can see, it seems like United assigns pilots on a purely volunteer basis, and even if they need to ask a pilot who had not signed up to perform a CRAF trip the pilot has a right to refuse. I couldn't find similar guidelines for other personnel needed to perform an airlift (flight attendants, fuel and baggage handlers, traffic control, etc.) but regardless of the specifics it seems that the airline has full control over the matter of personnel.