I would be careful of the use of "should". Here is a link to the regulation. I have not been able to find where it says that the Captain must verify this as it appears that it can be done by the gate agent.

Jumpseat Information > Home

8900.1 Vol. 3 Ch 2 Sec 1, Exemptions, Deviations, Waivers, and Authorizations

F. Identity and Eligibility Verification Methods. The following guidance is to be addressed in operators’ procedures in verifying an individual’s identity and jump seat eligibility. Methods are arranged by type of operation.

1) Method #1: Passenger-Carrying Operations—Employees of the Part 121 Certificated Air Carrier, Wholly Owned Domestic Subsidiaries, or Domestic Code-Share Partners. Certificate holder procedures for accessing the flight deck (jump seat) must include verification of identity, employment status, and jump seat eligibility at gate check-in using the procedures outlined in subparagraph 3-46D.

2) Method #2: All Operations—Eligible Employees of Other Part 121 Certificated Air Carriers. Certificate holder procedures for accessing the flight deck (jump seat) must include verification of identity, employment status, and jump seat eligibility at gate check-in using the procedures outlined in subparagraph 3-46D and as follows:

a) Additional items for non-company flightcrew members accessing the flight deck:

1. Current medical certificate if the requester serves as a flightcrew member for his or her employer.
2. Appropriate airman certificate.
b) Flight deck access by non-company employees must also be authorized by the issuance of OpSpec A048.