The most obvious way is to book and flight and hotel, or make a side trip while on vacation. What most people don't realize is that most offshore banking instituations will not open an account unless you are introduced to them by a reliable source. They will be polite and let you fill out an application, but the account will never actually open until someone vouches for you.
The best way to open an offshore banking account, is to use the offshore legal services of a law firm. For roughly the same price as your airfare and hotel bills you can open a Panama bank account through a lawyer.
Don't be suckered into a high price law firm who will charge $50k for what should cost less than 2k.
If you are not require to send your passport be weary. In todays post 9/11 world you will be required to provide documentation, even though the company and bank account are not in your name. The law firm must know who you are.