Query for array elements inside JSON[] field array type

Your problem stems from the incorrect use of the type json[]. A json array is a single json object and its type is json, not json[]. Example:

create table test (id int, crew json);
insert into test values
(1, '
[
    {
        "workHours": "9",
        "workers": "50",
        "checker_rate": 100,
        "rate": 150,
        "name": "Ramona",
        "last": null,
        "boxRate": 2,
        "checkTraining": false,
        "editing": true,
        "ix": 0,
        "breakPay": 3.0833333333333335,
        "trainingPay": 0
    },
    {
        "workHours": "4",
        "workers": "50",
        "checker_rate": 120,
        "rate": 160,
        "name": "Ramon",
        "last": "Rosas",
        "boxRate": 2,
        "checkTraining": false,
        "editing": false,
        "id": 1,
        "breakPay": 1.5416666666666667,
        "trainingPay": 0
    }
]');

The function json_array_elements() works as expected:

select id, elem->'name' as name
from test
cross join json_array_elements(crew) elem;

 id |   name   
----+----------
  1 | "Ramona"
  1 | "Ramon"
(2 rows)

One of the queries (or both) should work well with json[]:

select id, elem->'name' as name
from test
cross join json_array_elements(crew[1]) elem;

select id, elem->'name' as name
from test
cross join unnest(crew)
cross join json_array_elements(unnest) elem;