You'll probably be worried and paranoid on your first trip. But after a few experiences you'll learn that the world is not as dangerous as it seems.
Securing money and your passport
If you cannot secure your money and passport in a hotel safe or locker (e.g. when taking the train to next city) carry most of it on a hidden money belt. Carry the money (and maybe one card) one will need for the day in your wallet (in your front/zippered pocket) so you don't have to display the money belt. And hide some emergency money (and card) on your luggage.
The best place to keep any jewellery or the likes of anything with high financial or sentimental value is at home.
Avoiding mugging and violent crime
Check for safety warnings. The best information on safety is provided by the USA, UK, and Australia. They'll all say similar things. Remember if they advise 'do not travel' to your chosen destination, travel insurance will not cover you if you go there.
Violent crimes towards tourists are restricted to very few area, specially to places in South America and Africa. Consult your travel guide if any extra caution is needed for your destination.
Avoiding pick pocketing and casual theft
Check your travel guide for the scams practised on your destination. Or look at this list of common scams.
If you have been to the ATM or bank, drop it off or secure it first.
Don't flash cash around or leave your smart phone or tablet unattended. Don't carry something that looks like a handbag.
Remember the vast amount of crime happens at night (after the sun has set). Don't stray too far, get too drunk/stoned and return via taxi if necessary.
Trust your gut. If unsure of an area ask a local/hotel reception before leaving.