Stay closer to the MOA and International Airport (they're right next to each other)most definitely! It's cheaper and there are more family friendly hotels, than downtown Minneapolis.
However, if you're looking for luxury and/or nightlife, downtown might be more your cup of tea. There are several very nice hotels downtown as well as great restaurants, bars and clubs. But as I said, be ready to pay some serious buckage. It can be pretty busy and "crazy" at night in the area called block E, near where the stadium is located -- not necessarily family friendly, though you'll see alot of people with kids down there pretty late...especially leaving the game. So you might want to keep that in mind.
One of the main hubs for catching the light rail back to town from the ball game is next to some strip clubs and sometimes it can be interesting there for those waiting for a ride. If you rent a car, there is plenty of parking, but no guarantees you won't have a long hike to and from the stadium depending on where you wind up having to park. On weekends, parking is at a premium so plan accordingly and consider bringing along a light stroller if your kids don't like walking.
If you rent a car and drive, plan on getting downtown at least two hours early to get good parking close to the stadium, and plan on hanging out there or at a local restaurant. Event parking on the street or ramps can be $10 and up.
Taxis are plentiful after games and during after Happy Hour bar hours...so hailing one isn't difficult if you'd prefer a cab...but they are expensive if you leave the downtown area. It'll probably cost about $25 to $30 bucks one way to or from Bloomington to downtown Minneapolis. Alot of people hire a limo service for the night. It can cost about the same.
If you do opt to stay near the MOA, be careful to avoid the strip of hotels about a mile or two west from the airport and MOA, though. The area goes from decent to pretty rough, pretty quick along I-494... the closer you get to the Richfield border.
As others have said, there is the light rail and it will get you directly to the ball park and downtown from the Bloomington Strip (where the hotels, airport and mall are located).