A number of countries want you to have a certain amount of self financial capability before they will grant you the privilege of entering their country. This makes sense, these countries don't want people coming in and tapping their national resources. In New Zealand, for example, if you want a 6 month visa, you must be able to show you have $700 USD per month you want to stay. This can be demonstrated via bank statement.
We suspect that the formality around demonstrating the net worth part of the immigration policy is often left to the discretion of the local immigration official. That is, if we come in looking like we are financially sound (e.g. not coming in wearing tattered rags and appearing emaciated), we will be fine in most places. In policy driven environments, however, some form of proof must be on hand. This means that we will have to carry on our boat some proof of our financial soundness.
The trick will be to carry enough proof of financial self sufficiency to get in, but not so much as to then be susceptible to being taken advantage of (mysterious new fees). While we are modest in our means, everything is relative so modest in the US is wealthy in most parts of the world.
Our approach will be to have different bank statements. Since we have multiple accounts across multiple banks, we will simply keep the levels of money in each one different enough that we can produce the bank statement that sufficiently gets us into a place and no more.
Will this be a major issue? Probably not. However, we are thinking ahead and it is an issue one may face when taking a trip like this.