The Basic Authentication header encodes username and password. Effectively, it's just Base64 plus a "Basic" prefix.
You can use ActionController::HttpAuthentication::Basic.encode_credentials for that, and put its result into the Authorization request header.

Request specs

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For request specs, use the :header option.

it 'requires authentication' do
  get '/'
  expect(response.status).to eq(401)
end

it 'accepts valid credentials' do
  encoded_credentials = ActionController::HttpAuthentication::Basic.encode_credentials(username, password)
  get '/', header: { Authorization: encoded_credentials }
  expect(response.status).to eq(200)
end

Controller specs

In controller specs, you can put then into request.env['Authorization'].

it 'requires authentication' do
  get :index
  expect(response.status).to eq(401)
end

it 'accepts valid credentials' do
  encoded_credentials = ActionController::HttpAuthentication::Basic.encode_credentials(username, password)
  request.env['Authorization'] = encoded_credentials
  get :index
  expect(response.status).to eq(200)
end