I'LL BITE

The FRESHMEN dorms that are closest to CAP21 would be the two in/near Union Square....Third Avenue North (11th and 3rd) and University Hall (14th and 4th). U Hall is a little closer to CAP because it is right in Union Square but both are the closest to CAP of all the options. CAP is at 18th between 5th and 6th Avenues.
A couple of things.....first, the options for freshmen dorms are either traditional style dorms or apartment style dorms. It so happens that 3rd North and UHall are both the only apartment style dorms for freshmen. My D wanted apartment style. Basically, those two dorms have an apartment usually for four people....two bedrooms of two each, a living/dining room, kitchen and private bathroom. Traditional dorms for freshmen are located near Washington Square which is where the academic buildings are mostly near. The traditional style dorms at NYU still have private bathrooms but no living room or kitchen.
Some of this is personal preference but the Union Square area is a really cool place to live for these guys.
Your D would be at CAP three days per week for classes (though possibly at night for rehearsals if in a CAP production) and then two days per week would have classes closer to Washington Square or the Tisch building with is kinda between Wash Sq and Union Square. So, if she lives in a traditional dorm near Wash Sq, she is closer to classes on those days and if she lives in Union Square, she is closer on CAP days. The thing with Third North is that it is a few blocks south of Union Square and so maybe a little less far from academic classes than UHall but not that significant. Third North has its own dining hall. UHall uses the dining hall in the dorm adjacent to it. My D lived in Third North her freshman year. In her soph year, she lived in Union Square....even closer to CAP21 (could go home to her apartment for lunch) at Carlyle Court. She now lives off campus. Third North is a very big dorm (most NYU dorms are big but I think it may even be the biggest?) and it has a very nice interior court yard. Apartment style dorms cost more than traditional style dorms (hate to break that news).
Also, you have to put down like five or six choices of dorms for freshman year in order of preference. The ED frosh get first dibs, sorry. My D (who applied RD) road on the coattails of her roomie (theater friend she knew from our state for years) who was admitted ED to Tisch/CAP21 and so they got their first choice.