I don't recall any bars in NYC having a dress code, so T-Shirt and Jeans should be fine as long as you don't come in looking like a complete bum.
Having said that there are quite a few rooftop bars.
Here is the list of some of them that have menus available.
230
Fifth
The Heights
The Local cafe
But you can simply look at the top 10 list(?) to pick the one you fancy the most.
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The rooftop bar at Hotel Metro in midtown is unpretentious, cheap (by midtown standards), and gives you a pretty spectacular view of the Empire State building, though not a lot more (since you're more or less surrounded by tall buildings).
They have no dress code.
Top of the Strand (at the Strand Hotel) is similar to Hotel Metro - same neighborhood, same view, same prices.
The Brass Monkey in Chelsea is great - laid back, awesome views, very casual, and reasonably priced.
Other than that, my advice is to avoid going to any of the "top-rated" places (unless it's a list of the top 10 "unpretentious" rooftop bars like in Karlson's answer ;), as those will tend to be both pricey and snobby.
(FYI: $20 for a cocktail is definitely upscale pricing; mid-range bars will be more like $10-13 for a cocktail and $8-10 for beer.)
Using review sites like Yelp will give you a good idea of places convenient for you (NYC is very big!) and will almost always give you a heads-up about unspoken dress codes, rooftops closed for renovations or the off-season, etc.
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Here are two hotel bars where you can find quality cocktails (fresh ingredients, top shelf liquor, skilled bar staff): Soaked at the Mondrian Soho hotel - they have a great punch program Gallow Green is a rooftop bar at the McKittrick Hotel.
They have aserious classic cocktail program and occasionally the servers willtake on characters from a bygone era For a more casual atmosphere, try VU rooftop bar in Korea Town.
It's located in a random location, La Quinta Hotel, and offers a spectacular view of the Empire State Building.
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The Jimmy at the James Hotel in Soho is a beautiful place to get a drink.
So too is Le Bain atop the Standard in the Meatpacking District (probably my favorite rooftop in Manhattan).
But I say skip Manhattan and hop on the L to Williamsburg.
The Wythe Hotel has one of the best rooftop bars in the 5 boroughs.
Anyway, isn't the view of Manhattan better than the view from it?
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I realise this question is old but if anyone is looking for something a little unusual with great uninterrupted views of the park and midtown then
the Met have a roof garden/terrace which is open at certain times of the year (usually the beginning of summer, May/June).
For the cost of admission to the Met (suggested donation) there are reasonably priced drinks for Manhattan with great views out over the park from 5th Avenue.
No dress code required.
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Currently I'm in New York. I'm looking for a rooftop bar that provides a great overview about Manhattan, but is still affordable. (20 dollar for cocktail at maximum.) Additionally, I want to go there wearing jeans and a t-shirt, so nothing too fancy.
Rooftop bar in Manhattan?
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230, Fifth, The Heights, The Local café, the rooftop bar at Hotel Metro (with a great view of the Empire State Building), Top of the Strand, The Brass Monkey in Chelsea and The Jimmy at James Hotel are all good places to get a drink with a view without being too expensive. Bars in NYC generally don't have a dress code, so T-shirt and jeans should be fine as long as you aren't too scruffy. New York is full of cocktail bars and many have a great view , including The Heights and The Brass Monkey. Two bars are expecially recommended for quality cocktails and they are Soaked and Gallow Green.
Bars in New York do not usually have a dress code. There are many good bars in New York for cocktails with a view, including The Heights, The Brass Monkey, Top of the Strand, among others. Two bars are recommended for quality cocktails. They are Soaked and Gallow Green.