Almost the entire crypto-exchanges, new or old, already experienced several problems on their way. But to give you some "old" exchanges that are still operational as of today, there are; (Not sure about the specific meaning of "old" so I will assume been in service for over half a decade, at least)

- Poloniex
- Bitfinex
- Bitstamp
- Coinbase

Though I don't know some "old" exchanges that operated during the MT Gox period that still exist today.