First of all, it must be stated that scientists draw no real
distinction between hair and fur.
See: "What is the difference between hair and fur?" Scientific American,
Or the same article stored by archive.org
Ask a Scientist! "What is the difference between hair and fur?"
Moment of Science: The Hairy Truth
Now, getting back to your question...
Monkeys do have "fur." Numerous sources refer to monkey hair as "fur."
"Spider Monkeys... Their fur color can be black, brown, golden, red, or tan."
"Douroucoulis, or Night Monkeys... The body is so heavily covered with
fur that it appears to be short and thick."
source: Mammals of the World
"...capuchin monkeys... rub their fur with crushed millipedes"
source: The Primates: New World Monkeys
Nonetheless, as mentioned above, it's important to remember that
monkey fur is hair.
Here's an enlarged image of a monkey hair:
From an interesting web site that compares the hair/fur of many
mammals Hairs Polarized Light Digital Image Gallery, hosted by Florida
"between hair and fur"
"between fur and hair"
"monkeys have fur"
I hope this helps.