In the old days, when Las Vegas hotels were smaller and more intimate, many featured balconies that looked out onto the Strip or the pool area. But today, few Strip rooms have outdoor access and we can think of a number of reasons for this.
For one thing, Nevada has one of the highest suicide rates in the nation and jumping (be it from the Stratosphere Tower, the Hoover Dam, rooftop parking lots, overpasses, or hotel balconies) is a popular choice. Hence, most hotels don't even have opening windows, let alone balconies. Another reason is almost certainly the climate -- there are many months of the year when being outside is not a desirable option -- and this ties in with a third reason, namely the additional expense both of building balconies and of air-conditioning a hotel with open windows.
That said, some properties do offer this amenity, although few of them afford views of the Strip. And a word of caution: Several hotels that we called stated that they had suites with "balcony views," which as far as we can tell simply means that they have large windows! As far as those with actual outside access are concerned, here are the options:
Harrah's: A select number of rooms overlooking the pool have small balconies i.e., not large enough to take a chair onto; we've seen them described as "a really narrow concrete ledge with a railing." We read one trip report where a couple said their "balcony" gave them a view of the Venetian, TI, and the mountains, while another complained that theirs "opened out onto a bird's eye view of a giant fan."
Imperial Palace: Penthouses have full-length patios; some of the deluxe rooms have balconies with a Strip view, but a balcony is not guaranteed when you book.
MGM Grand: Patios are available in the Terrace Suites (total of 14) -- 1,300-sq.-ft., two-story rooms with 800-sq.-ft. patios complete with hot tubs. These have views looking out toward New York-New York and Excalibur or the Monte Carlo.
Riviera: They have five suites, most or all of which (we couldn't get a definitive answer on this) have balconies and range from one-room-studio style to two-bedroom penthouse. The suites are available to the public, subject to availability, and rooms either overlook the pool or the Strip.
Royal: Some rooms (those with even numbers) have balconies, but they have only north- and south-facing views, i.e., not of the Strip.
Sahara: Some rooms and suites have balconies at no extra charge, but they cannot be guaranteed.
San Remo: The Gold Tower rooms currently have balconies but they may be done away with in the future. Views are of the MGM Grand or of the Strip.
Tropicana: About two-thirds of the Garden Rooms have balconies, but these are located in the low-rise buildings and thus offer little in the way of views, plus they cannot be guaranteed.