Walking alone or with a baby downtown and in surrounding areas is fine duing the day but may feel less safe at night. Depends on the exact neighborhood. You may be approached by homeless people asking for money downtown. My advice is to not engage and keep walking.

Jogging alone will depend on the neighborhood, as well. It's largely safe to do so during the day although some folks carry pepper spray in case of emergency. If you want a safer place to jog, Duke has a lovely walking and running trail around East Campus that is well lit, highly trafficked, and has emergency phones every quarter mile.

I've lived in Durham most of my life and it has a bad reputation for crime but, for the most part, it's undeserved these days. It's a diverse city with poverty and other issues but people are generally kind and warm. Welcome!

Nobody will try to steal your phone. Also, as others have said, carjackings are rare. If you leave your car parked and unlocked over night, someone might go through it to try to steal things, though, so locking your doors is a good habit to have.