Question

What is the best way to punctuate the bolded portion of the sentence?

Answer 1

It would be punctuated like this: The Atlantic Monthly.

You would not put it in quotation marks. That is for smaller literary works. A magazine title would be in italics only. Here's a few sources supporting this.

http://www.thepunctuationguide.com/titles-of-works.html

https://allaboutwritingconsulting.com/2013/12/how-to-punctuate-titles/

Answer 2

The best way to punctuate the Atlantic Monthly is "The Atlantic Monthly." But you would also italicize it because Its a title.                                                          Hope that helps! :D