What is shomer enayim? I would hardly call Ner Yisroel “modern”, I think “less Yeshivish” is a better term. Both NIRC and Schlangers (Beis Medrash and Mesivta of Baltimore) are very good Yeshivas but they are also very different type of places and what is good for the goose may not also be good for the gander.
Both Yeshivas require hats during davening, on Shabbos and during the week. Schlangers requires only white shirts. But if you’re picking a Yeshiva for your son based on what the dress code is, may I suggest, well, not doing that?
NIRC is a big place. First year beis medrash has something like forty bachurim and two rebbeim. And the numbers are similar for all the other years up until sixth where everyone starts getting married and moving to kollel. First thing you should know is that there are several different Ner Yisroels in the same building. There are the guys who came straight from the Mechina (high school) and generally stay for a year or two before going to Eretz Yisroel and then a different American Yeshiva. There are the Persians and Latinos who often come with no Yeshiva experience and group together in some of the “easier” shiurim for the first few years. There are the guys who come when they are 21 just because they want a Yeshiva that allows college and they don’t want to go to Touro or YU. And so on.
Schlangers is a lot smaller. I believe that there are only about 80 to a hundred bachurim there. The bachurim are a lot more integrated and close with each other. Most of them were there for Mesivta (high school. Seriously, can Yeshivas just decide on one word to use for high school and stick with it?) and decided to stay on. It’s much more Yeshivish in the sense that there is an enforced dress code and no college allowed.