I think St. Mary's hospital has parking for $10 behind the strip. There is free parking in the Market Square and State Street parking garages. There is a shuttle from Market Square, but it's $5 a person, so it would be cost effective to just pay $20 to park nearer the stadium.
If your kids are old enough that they won't complain about the walk, I just walk from State Street garage. It's really not that bad a walk.