As a pp said, Jet Blue requires a photo ID for those 18 years and older.
The reason you would need the birth certificate is for proof of age (to prove that a lap child is indeed under the age of two)...and not identification.
For identification purposes you would need a picture ID so the childrens' birth certificate will do you no good anyway.
From the JetBlue website:
ID Requirements for Domestic Travel
Customers over the age of 18 will be required to show a U.S. federal or state-issued photo ID that contains the following information: name, date of birth, gender, expiration date and a tamper-resistant feature. These IDs include:
U.S. passport
State-issued driver's license
U.S. passport card
DHS "Trusted Traveler" cards (NEXUS, SENTRI, FAST)
U.S. Military ID
Permanent Resident Card
Border Crossing Card
DHS-designated enhanced driver's license
A Native American Tribal Photo ID
An airline or airport-issued ID (if issued under a TSA-approved security plan)
A Registered Traveler Card (that contains the following: Name; Date of Birth; Gender; Expiration date; and a Tamper-resistant feature)
A foreign government-issued passport
Canadian provincial driver's license or Indian and Northern Affairs Canada (INAC) card
Citizens of countries other than the US or Canada are NOT required to carry their passports if they have documents issued by the U.S. government such as Permanent Resident Card. Those who do not should be carrying their passports while visiting the U.S.
The name on the ID must match the name in the reservation.
Infants/Lap Child
Customers traveling with an infant as a lap child may be asked for proof of age, such as a passport, birth certificate (copies are acceptable for domestic travel) or an immunization record when the infant's age is questionable.
Infants between three and 14 days old must also have, in the form of a letter, their doctor's approval to travel.
Unaccompanied Minors
Unaccompanied minors must check in at the ticket counter. An unaccompanied minor does not need photo ID. However, the adult dropping them off and picking them up will need photo ID. This adult must be named in the reservation.
Have a great trip.