Hello,
ATMs in Ecuador charge the same fees everywhere, all banks are reliable and they biggest one that you will find everywhere is Banco del Pichincha. In Galapagos I believe you also can find Banco del Pacífico.
When you use a debit card, you will probably be charged a fee of USd 0,50 or USD 1,00 per transaction and there is a limit of withdraw, depending on your bank policies. Other additional fees may be charged by your bank due to international use.
When using a credit card, I believe you will be charged with fees of your bank, due to international transactions. Please check with your local bank.
It is safe to use any ATM in Ecuador, however, you need to be careful for the people around you and "friendly" guys who can come to you to help you with the use of the ATM. You will need to take the same precautions you will take in a big city. Try not to go at night.
Edited: 6 years ago