Driving city buses is quite different from driving intercity/long haul buses, and in this last category we include airport buses.
City bus drivers work 12 to 17 hours a day, under very stressful conditions. Some long haul companies also press their drivers too far. Most suffer from kidney stones from lack of "evacuation" facilities and uneven eating schedules.
Airport buses are mostly reputable and reliable. You have the option of Guo Kuang Hao, EVERGREEN Shuttle, plus a myriad of hotel airport shuttles also available to the general public. I haven't heard of any of these airport buses catching fire or having a strepitous accident as often as other long haul buses do.
Top rate to the airport is like 1100 ntd by taxi, which is quite reasonable given the distance, IMHO. If that is not good, you have provate drivers competing with fares as low as 900 ntd for a van, for example.
Taxi drivers that belong to companies, like Taiwan Taxi, Ta Fong, Metro, etc. are usually quite professional in their driving and must upkeep their carsproperly. It is the independents that show up with pinglan and karaoke and other less amenable "amenities". You, as a customer, can make the informed hoice. You do not have to take the first one that comes along. Take the bad ones off the market by not giving them your patronage. If you have any problem with a taxi that belongs to a company, you have ample ways to complain and be heard. Taxis here are clearly identified.