Don't smile, don't say anything, don't stop. Just keep walking. Duck into a shop if a vagrant or anyone else seems to be trailing you. Also, the parks are generally safe during the daylight hours, but again, try not to venture out alone. In most studios, you can put your dance bag under the piano or near a barre during class.
Some studios also provide racks and hangers in the classrooms.
In coffee shops and restaurants, put your dance bag under your chair or stool, between your legs. Seasoned New Yorkers automatically put one leg through the strap of their bags, just to be sure. Lock both, even if you're just going down the hall to the garbage chute. Enroll in a card protection plan if possible, but at least write all numbers down and put them in a safe place in case your wallet is stolen. Vagrants make a practice of watching people punch in calling card numbers.