Very important every decent charting site uses an API, bitcoinwisdom.com cryptowat.ch tradingview for instance. So even if you don't directly query the api, you need it to functionally evaluate price as internal charts on exchanges are pretty bad.

As far as "people" that actually use APIs you're looking mostly at bots or people gathering data for backtesting purposes. There are some things like QT-Trader, cryptrader, coinigy which by adding your api keys you can view and submit orders.

tl;dr I wouldn't run an exchange without an API.