I carry my passport with me when I am taking my own excursions and tours.
Is it really needed? I have only had to show it a couple of times when using my credit card and in places like China and Russia it is a requirement.
The choice is yours but I would definitely take it in Croatia.
Like Kuki said, use bank ATM's and do not use the sidewalk no-name ATM's. I also do not use travelers checks. Get some Euros when you land at the airport and if you need more just get them at the ports.