Are there machine learning methods that can give captions to the New Yorker magazine cartoons? 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.