Ok let me start of with the following, Moncton is a Rather small city therefor there are no Red Zones or area with very high crime rate as you will find in Compton, California for exemple.
That being said, there are still area that should should avoid, and those are Close to downtown in the square of King St, Queen St, and Botsford St. Now having said that you are not going to be in great danger but it is a residential area where incomes are low and overall the area is a little run down. Victoria Park Between Cameron and Weldon is known for Prostitution and Drug trafficking or use. There should not be a problem in the day time but at night I would not advise going there.
The area of between Beachwood Ave, McLaughlin Dr, VAil and Elmwood are a little shady therefore should probably be avoided, police are often called in for different thing. There are nothing to see there anyway so you won't be missing anything.
Downtown in General is Very safe and can be somewhat busy for clubs and restaurant during weekends, there are usually police patrolling on foot so if something should be to happen, should not last very long.
There are cases of Mugging and Stabbing occasionally around the city, most of them happen at night time, if you need to travel at nighttime take your car or you can always call a taxi.
Beggars tend to stay around Main st, Downtown area, and will ask for money, simply continue walking or say don't have anything. They are trying to get them away from the core so if one gets aggressive just threaten to call the police.
As in every city, Lock your cars doors and put valuable out of sight.
Enjoy your visit :)
There's only one transient that seems to be a handful. He squats at a keyboard and bangs out noise while shouting what I believe are obscenties in Patoois. Hard to say. He is tempermental and irratic. Occasionally his outbursts make the newspaper, but I missed the article. I will post a picture. Best to simply leave him alone and try not to stare. He said I could take his picture, but then covered his face.
I once walked through downtown Moncton with a black friend of mine. I've never seen one guy get so many looks in my life. You'd swear this guy was the first black guy the town has ever seen. This one Eminem look-a-like came up to my friend, shook his hand and said "Respect, Whut up mah brotha, welcome to Moncton". The local Moncton Bum even showed his respect by coming over to my buddy and giving him a hug. Be prepared for overwhelming respect if you're a black guy visiting moncton.
Moncton winters are extremely cold, so make sure you dress warmly. It is not entirely unusual for temperatures to drop to about 40 below. Also, Moncton gets a lot of snow, so make sure you have high boots. Blizzards can occur without warning, sometimes as late as May or early June.