I've been traveling with my wife and two kids for 3 1/2 months already for less than your budget. Never felt unsafe anywhere. Thailand, Malaysia, Israel, Jordan, Greece, and now Italy. My advice...cheaper parts of the world use less of your budget on lodging/transport so you feel free to spend on food and experiences. Way more bang for buck.
All that said, if you are worried about safety (turn off the tv!) and travel convenience matters for your timeline or something it is hard to beat Kauai. We love it there. Renting a condo can be nicer and cheaper than the resorts.