The Gateway, which is a ritzy shopping center containing the Union Pacific station is directly north of a homeless shelter. Do not stay around there.
Although the cops and city regularly try to clean up Pioneer Park, it is still the place you do not want to go. Homeless and druggies abound.
The Main Street Crossing between the two halves of City Creek Center usually have a couple of panhandlers, but I haven't found them to be very obnoxious. Temple square sometimes has a few panhandlers or Bible thumpers being annoying OUTSIDE on the sidewalk. They aren't allowed inside, and LDS church security, although you won't see them, is always present.
I would say it's mostly tourists being extra nervous. After going to other cities like San Francisco, the panhandlers and homeless aren't that big of a deal. I've never felt unsafe in any parts of Salt Lake City.
But I would still completely avoid Pioneer Park. There's nothing good to see there. Only trouble.