Are there machine learning methods that can give captions to the New Yorker magazine cartoons?
Yes. Image captioning is a very active area of Deep Learning, at the intersection between Computer Vision and Natural Language Processing.

These models are based on an Encoder-Decoder structure. In its most classical form, the Encoder consists of Convolutional layers that process pixel data. Their representation is then fed to a Decoder based on Recurrent layers (usually LSTM or GRU) that generate the output sentence.

I suggest you to take a look at this tutorial on Image Captioning in TensorFlow 2. You can download the Notebook and run. That specific model is based on attention mechanism, which is a fancier version of seq2seq models. You can find tons of other models and tutorial simply googling them.