How is the currency of Taiwan referred to in Mandarin?
Yes, people use ‘kuài’ in conversation, as in ‘yī qiān duō kuài’ (over 1,000 NT$).  You can also add ‘qián’ to make it clear you’re talking about amounts of money:  ‘wŭ shí kuài qián’ (50 NT$).

You might want to use ‘(xīn) tái bì’ when changing money, as in ‘qĭng gĕi wŏ tái bì’ (please give me Taiwan dollars).  I don’t know what was used in previous periods, but here are some ways to refer to other currencies:


mĕi jīn   ‘American dollars’ 
rì bì   ‘Japanese yen’
găng bì  ‘Hong Kong dollars’
yīng bàng  ‘British pounds’


These are from a Mandarin textbook put out by TLI, a language school in Taiwan.  Be careful with the tone on ‘tái’ - in some contexts this can be confused with the falling tone ‘tài’ referring to Thailand/Thai money.  But if you’re already in Taiwan, that shouldn’t be a problem.