This is difficult to answer because Keyes was fairly conscientious about avoiding settings that would immediately place the novel in time. For example, there is no instance of hearing a particular pop music hit that might indicate the specific time period. However, the reader can catch glimpse of the specific setting through the technology and scientific concepts that are bandied about. Additionally, the character of Faye gives some clues but only if the reader is familiar with New York City in the late 50s and 60s.
For example, we know that 'learns' at night with the aide of a tape recorded lessons. This can set the time period as probably before the advent of CDs or MP3s.
The character of Faye living as an artist in New York City and as a fairly liberated woman gives us a clue to the time period of the late Beat Generation to the middle of the Sixties. Her opposite, Alice is a teacher in a night school working a fairly conventional job for an unmarried woman at the time period.
These are just some clues as to the setting. The 1969 film version of the movie gives the viewer very distinct clues as to the time period which set it firmly in the late 1960s. So much so that when I showed it to my students, they laughed at everything from the wacky '60s decor to the clothes the characters had on. The book does not do this and therefore it is hard to place it specifically in a time period unless you are familiar with the time.