The Stanislaus River park system provides multiple areas for adventurers to camp. The main areas are at Horseshoe Park, Valley Oak Recreation Area, and McHenry Recreation Area. All of the parks on the Stanislaus River Park system can be reserved by calling the Army Corp of Engineers at 209-881-3517.
All of the park campgrounds are well maintained by the Army Corp, and provide access to restroom facilities (but not showers). Rates vary based on group size, but are general much more affordable than commercially based camping options in the area (ex: group rates for groups of ~25 go for around $40-$50 per night total for the group). To make reservations you do need to book at least 2 weeks in advance (Army Corp rule), and there are parking restrictions, so please try to carpool to the camp site area.
Horseshoe Park and Valley Oak Recreation Area are, compared to a lot of other parks in Northern CA, incredibly quiet for riverside camping. There are quiet hours (10PM to 6AM), but in these areas, campgrounds are rarely packed, except for on holiday weekends). McHenry Rec tends to be a little bit more crowded during the summer months, but still provides a wonderful value for the accommodations provided.
For more specific information about the campgrounds, you can visit the specific park pages. For additionally information on general camping, check out the Army Corp Camping Rules Here.