Sewer alligator stories date back to the late 1920s and early 1930s, in most instances they are part of contemporary legend. They are based upon reports of alligator sightings in rather unorthodox locations, in particular New York City.
Following the reports of sewer alligators in the 1930s, the story has built up over the decades and become more of a contemporary legend. Many have even questioned the extent of truth in the original stories, some even suggesting it to be fiction and that Teddy May's creative mind may have contributed to the tales. However, the story of the 'Sewer Gator' in New York City is well known and various versions have been told.
The original story was that wealthy families would return from vacation from Florida and ignore the laws in New York City, bringing alligators with them, as pet presents to their children. The time frame of this tradition is rather gray, but it has been suggested it originated in the late 1930s. When the alligators grew too large for comfort, the family would proceed to flush the reptiles down the toilet (although a 'gator small enough to be flushed could hardly be considered "too large for comfort"). 
What happens next varies. The most common story is that the alligators survive and reside within the sewer and reproduce, surviving by feeding on rats and rubbish, growing to huge sizes and striking fear into sewer workers. In Robert Daley's book, 'The World Beneath the City'(1959), he comments that one night a sewer worker in New York City was shocked to find a large alligator swimming toward him. Weeks of hunting followed.
The Journal of American Folklore has this to say on the subject of May, 'The World Beneath the City' and Alligators in the Sewers:
In 1959 a book entitled The World Beneath the City was published by Lippincott. Written by Robert Daley, it is a history of the problems involved in the development of the network of utilities Manhattan Island. And in the midst of the stories of engineering problems and political deals is a chapter entitled "Alligators in the Sewers" (see pp. 187-189). It is based on the author's interviews with Teddy May, who had been Commissioner of Sewers in New York for some thirty years.
According to May, sewer inspectors first reported seeing alligators in 1935, but neither May nor anyone else believed them. "Instead, he set men to watch the sewer walkers to find out how they were obtaining whisky down in the pipes." Persistent reports, however, perhaps including the newspaper item discovered by Coleman, caused May to go down to find out for himself. He found that the reports were true. "The beam of his own flashlight had spotted alligators whose length, on the average, was about two feet."
May started an extermination campaign, using poisoned bait followed by flooding of the side tunnels to flush the beasts out into the major arteries where hunters with .22 rifles were waiting. He announced in 1937 that the 'gators were gone. Reported sightings in 1948 and 1966 were not confirmed.
However, there is no mention of "blind, albino" alligators, and May suggests that the baby alligators were dumped down storm drains rather than "flushed down the toilet."
An additional reference to the sewer alligator exists in Thomas Pynchon's first novel, V. (pp. 43). It fictionalizes the account, stating Macy's was selling them for a time for 50 cents. Eventually the children became bored with the pets, setting them loose in the streets as well as flushing them into the sewers. Rather than poison, shotguns were used as the remedy. Benny Profane, one of the main characters in the book, continues to hunt them as a full time job until the population is reduced.
Some versions go further to suggest that, after the alligator was disposed of at such a young age, it would live the majority of its life in an environment not exposed to sunlight, and thus it would apparently in time lose its eyesight and the pigment in its hide and that the reptile would grow to be completely albino, pure white in color with red eyes. Another reason why an albino alligator would retreat to an underground sewer is because of its vulnerability to the sun in the wild, as there is no dark pigment in the creature's skin, it has no protection from the sun, which makes it very hard for it to survive in the wild. 
The albino alligator, which does in fact exist, has rarely ever been sighted in the wild. Because of its color, the bright white and pinkish skin makes it vulnerable to predators as an infant as well as an obvious sight for any source of food it may attempt to collect..
There have never been any official sightings of this kind of alligator in New York City..
However, most experts believe that a sewer is not a fit environment for any alligator, and they would be unlikely to be able to reproduce. The animals need warm temperatures all year round. Of course, the major assumption behind these assertions is that the alligators in question would never venture back to the surface again, choosing instead to occupy the sewer/storm drain environment exclusively, which is actually rather warm year round.
The majority of New York sightings mention alligators that are small in size. Therefore they would possibly be caimans, a South American member of the family Alligatoridae which were once heavily imported for the pet trade and do not possess much of a tolerance for low temperatures. 
The TV series Monster Quest investigated this and discovered that the sewers may possibly be an ideal habitat for Alligators, as various heat source keep it warm enough and there are plenty of food sources and materials which could be made into a nest. Though they failed to locate a gator, they did discover a salamander, which was equally as unlikely to be found there.