I've been using my debit card for most transactions for the past 10 years or so. I cut up the credit cards along time ago and only have one I use for rental cars and hotels, both of those put ridiculous holds on your cards so its just more convenient that way. I've never had my info stolen, not once.
Best way to budget is with cash though, you never have any unexpected surprises, using your debit or credit card just leaves too much temptation to over spend especially if you're somewhere where everything ends in a gift shop.
That being said, if you have a bank that doesn't stand behind your debit card immediately and consistently you need a new bank. I was an hour from my home at a gas station and they called me right after I had purchased gas thinking it was a possible fraudulent transaction and were verifying it less than 10 mins after it happened.
If you are going to use your credit/debit card make sure you inform the issuing bank, actually talk to someone, no e-mails, online questions , etc.. Let them know the exact travel dates and locations. If you are on vacation and need to make a large purchase call before someone tries to ring it up.
Me personally, I do cash when on vacation. Cash stays on me on the way to where I am going, wallet in the front pocket of cargo shorts or jeans , also keeps my tush from falling asleep when sitting for long amounts of time. When I get to the hotel if there is a safe I will use it, if not I will take out what I have budgeted for the day and the rest gets hidden, I usually carry a garbage bag for dirty laundry and no one in their right mind is going to want to dig through my dirty socks buried in the bottom of a suitcase to find money. Make sure your house keeper doesn't have to move your luggage either, put it in a place where they can do their thing and have no excuse to touch your bag. I also do the laptop or other valuables in the suitcase with the suitcase stored as most hotel safes don't fit a laptop. I also make my screen saver a big "this laptop is the property of ...." when traveling too.
Feel safe to use cash, its the easiest way to keep overspending and teaches the little ones good habits. You don't end up buying that extra Mickey bar or cool Doctor Who Sonic Screwdriver pen when you know you won't have money for dinner later if you do. Be smart about it too, ladies keep the cash on you and not in a purse, guys wallets in the front pockets the back ones are to tempting of targets. Same goes for Credit Cards and ID's as well!