After the Chinese took over Tibet, many Tibetans came to live in exile in India. Darjeeling became the base for many of them - others went to live in McLeod-Ganj / Dharamsala, the home of the Dalai Lama. Monasticism has always been an important part of Tibetan Buddhism, and there are now many monasteries, and some nunneries, around Darjeeling.
My boyfriend spent a year in India between school and university, teaching English in Guru Sakya monastery, in Ghoom. He was teaching the baby monks - those aged between 7 and 18. Many boys go to the monasteries for an education, but not all of them go on to take vows later in life. The village is about 5 miles down the road to Calcutta from Darjeeling - but no one told him before he went that the name of the village was Nepali for "damp"....
Visitors are welcome at the monasteries, during the day, provided they behave appropriately. We went and visited a couple around the town, and were shown the prayer halls, wheels, and monastic buildings. It's a very interesting experience, and the monks are very kind and helpful. You may find that in the smaller establishments, the level of English is minimal, however.
A small donation, while not required, is gratefully received. I know from Adam's living in Guru Sakya monastery, that many are very poor indeed, to the extent of food shortages every so often. Many of the places sell prayer scarves - those long, white silky affairs used in meditative exercises, if you don't want to make a gift.