I don't think you'll have an issue with safety in Maple Grove. I've never heard of any problems there. As far as finding an affordable apartment that accepts dogs, I would suggest trying
craigslist: minneapolis / st paul classifieds for jobs, apartments, personals, for sale, services, community, and events and, under Housing, click "apts/housing." When the screen pops up, use the search box at the top to find a place in your price range and that accepts dogs. Just beware of people asking for your personal information BEFORE you see the apartment! Protect yourself from scammers. They legally cannot ask you for this information before you view the apartment, otherwise it's discrimination and pre-screening.
Otherwise, you can work with Apartment Search. They're a free service that helps people find places to live, according to your criteria.
Minneapolis Apartment Search: Find Apartments to Rent in Minneapolis, Minnesota I've used them before and they are VERY helpful... not to mention free. And, if you rent a place you find through them, they'll give you $100 bucks.
Other than those options, it's a matter of going through the rental magazines from the area, doing online searches or driving around and checking it out.
Best wishes to you!