Indiana Dunes State Park is about a mile from the Southshore Line that will take you downtown. Pretty reasonable to ride the train, staying at IDSP it's pricey, as they will peg you for being an out of stater.
Indiana Dunes National Lakeshore Campground is nice, but it's rustic (no electric at sites) they do have pay showers. It also is easy to get to the train.
These are ones off the top of my head, if I think of more I'll stop back!!
2002 Chevy K2500, Duramax, 312,000 miles & going strong
2015 Palomino Backpack 1251
(who puts the sewer drain on the right side??
Torklift tie downs w/ Fastguns