Why do banks engage in offshore banking?

Mostly to create a product offering that is not hinged on the laws of a specific country or a certain geographic region.

Many island nations had no mechanism of generating income other than tourism, farming (pineapple, sugarcane, etc.) or fishing. Other than these three verticals, there wasn’t an industry per se.

However, what they did have was sovereignty as a country and they were pretty much free to make their rules & laws. Neighboring countries first established Merchant Banks along these island nations. Mostly in line with the ports of call for merchant shipping.

Eventually, banks discovered mechanism to give a little bit of a fees, keep a few people employed and a steady stream of customers who much visit these islands to open accounts or do other activity, resulted in tourism related income. Lo and behold, offshore banking arose, which basically was offering comparable banking services except it was outside the tax limits of say the United States or United Kingdom or The Netherlands, etc.

Clients could park their funds here without the fear of being chased-down by the tax man. Also this money is then safely invested in bonds and other low-risk investments on-shore with banks that provide correspondent banking to these off-shore banks.