How to send custom headers with requests in Swagger UI?

You can add a header parameter to your request, and Swagger-UI will show it as an editable text box:

swagger: "2.0"
info:
  version: 1.0.0
  title: TaxBlaster
host: taxblaster.com
basePath: /api
schemes:
- http

paths:

  /taxFilings/{id}:

    get:
      parameters:
      - name: id
        in: path
        description: ID of the requested TaxFiling
        required: true
        type: string
      - name: auth
        in: header
        description: an authorization header
        required: true
        type: string
      responses:
        200:
          description: Successful response, with a representation of the Tax Filing.
          schema:
            $ref: "#/definitions/TaxFilingObject"
        404:
          description: The requested tax filing was not found.

definitions:
  TaxFilingObject:
    type: object
    description: An individual Tax Filing record.
    properties:
      filingID:
        type: string
      year:
        type: string
      period:
        type: integer
      currency:
        type: string
      taxpayer:
        type: object

You can also add a security definition with type apiKey:

swagger: "2.0"
info:
  version: 1.0.0
  title: TaxBlaster
host: taxblaster.com
basePath: /api
schemes:
- http

securityDefinitions:
  api_key:
    type: apiKey
    name: api_key
    in: header
    description: Requests should pass an api_key header.

security: 
 - api_key: []

paths:

  /taxFilings/{id}:

    get:
      parameters:
      - name: id
        in: path
        description: ID of the requested TaxFiling
        required: true
        type: string

      responses:
        200:
          description: Successful response, with a representation of the Tax Filing.
          schema:
            $ref: "#/definitions/TaxFilingObject"
        404:
          description: The requested tax filing was not found.

definitions:
  TaxFilingObject:
    type: object
    description: An individual Tax Filing record.
    properties:
      filingID:
        type: string
      year:
        type: string
      period:
        type: integer
      currency:
        type: string
      taxpayer:
        type: object

The securityDefinitions object defines security schemes.

The security object (called “security requirements” in Swagger–OpenAPI), applies a security scheme to a given context. In our case, we’re applying it to the entire API by declaring the security requirement a top level. We can optionally override it within individual path items and/or methods.

This would be the preferred way to specify your security scheme; and it replaces the header parameter from the first example. Unfortunately, Swagger-UI doesn’t offer a text box to control this parameter, at least in my testing so far.